Periodic motions and limit cycles of linear cable galloping

Abstract

In this paper, analytical galloping dynamics of linear cables with the uniform airflow of the wind is discussed through a two-degree-of-freedom nonlinear oscillator. The nonlinearity in the two-degree-of-freedom oscillator is only from aero-dynamic forces caused by the uniform airflow. The analytical solutions of periodic motions for linear cable galloping are discussed from the analytical method with the finite Fourier series. The corresponding stability and bifurcation of the periodic motions of the linear cable galloping are determined through the two-degree-of-freedom nonlinear system. Frequency-amplitude analysis of periodic motions and limit cycles of linear cable galloping are presented. Numerical illustrations of trajectories and amplitude spectrums are given for galloping motions of linear cables. From such analytical solutions, galloping phenomenon in flow-induced vibration can be further understood. The galloping dynamics of linear cables is similar to the dynamics of the van der Pol oscillator.

Appendices

Appendix: Coefficients for cable galloping

In Appendix, coefficients for galloping cables are presented. Such coefficients are suitable for all fluid-induced vibration through the two-degrees-of-freedom oscillators.

1.1 Coefficients for Fourier integrations

The basic quantity variables are defined to simplify the complicated expressions in coefficients for nonlinear terms.

$$\begin{aligned} B_{k/m}= & {} \dot{{b}}_{2k/m} +\frac{k\Omega }{m}c_{2k/m} ,\;C_{k/m} ={\dot{c}}_{2k/m} -\frac{k\Omega }{m}b_{2k/m} ,\; \nonumber \\ P_{k/m}= & {} \dot{{b}}_{1k/m} +\frac{k\Omega }{m}c_{1k/m} ,\;Q_{k/m} ={\dot{c}}_{1k/m} -\frac{k\Omega }{m}b_{1k/m}.\nonumber \\ \end{aligned}$$

(59)

The delta functions for constant terms are

$$\begin{aligned} \Delta _1 \left( i,j,l\right)= & {} \delta _{j+l}^i +\delta _{l+i}^j +\delta _{i+j}^l , \nonumber \\ \Delta _2 \left( i,j,l\right)= & {} \delta _{j+l}^i +\delta _{l+i}^j -\delta _{i+j}^l , \nonumber \\ \Delta _2 \left( i,l,j\right)= & {} \delta _{j+l}^i +\delta _{i+j}^l -\delta _{l+i}^j , \nonumber \\ \Delta _2 \left( l,j,i\right)= & {} \delta _{l+i}^j +\delta _{i+j}^l -\delta _{j+l}^i . \end{aligned}$$

(60)

For the constant terms of nonlinear functions, we have

$$\begin{aligned} f_1^{(0)}= & {} \big ({\dot{a}}_{20}^{(m)} \big )^{3}+\frac{3}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \big (B_{i/m}^2 +C_{i/m}^2 \big )\nonumber \\&+\,\frac{1}{4} \sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3C_{i/m} C_{j/m} B_{l/m} \Delta _2 \left( i,j,l\right) \right. \nonumber \\&\left. +\,B_{i/m} B_{j/m} B_{l/m} \Delta _1 \left( i,j,l\right) \right] , \end{aligned}$$

(61)

$$\begin{aligned} f_2^{(0)}= & {} \big ({\dot{a}}_{20}^{(m)} \big )^{2}{\dot{a}}_{10}^{(m)} +{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m} \right) \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\big (B_{i/m}^2 +C_{i/m}^2 \big )} \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ B_{i/m} B_{j/m} P_{l/m} \Delta _1 \left( i,j,l\right) \right. \nonumber \\&+\,C_{i/m} C_{j/m} P_{l/m} \Delta _2 \left( i,j,l\right) \nonumber \\&\left. +\,2B_{i/m} C_{j/m} Q_{l/m} \Delta _2 \left( l,j,i\right) \right] , \end{aligned}$$

(62)

$$\begin{aligned} f_3^{(0)}= & {} a_{20}^{(m)}\big ({\dot{a}}_{20}^{(m)} \big )^{2}+\frac{1}{2}a_{10}^{(m)} \sum _{i=1}^N \big (B_{i/m}^2 + C_{i/m}^2 \big )\nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) } \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} B_{j/m} B_{l/m} \Delta _2 \left( i,j,l\right) \right. \nonumber \\&+\,b_{2i/m} C_{j/m} C_{l/m} \Delta _2 \left( l,j,i\right) \nonumber \\&\left. +\,2c_{(2)i/m} B_{j/m} C_{l/m} \Delta _2 \left( i,l,j\right) \right] , \end{aligned}$$

(63)

$$\begin{aligned} f_4^{(0)}= & {} {\dot{a}}_{20}^{(m)} \big ({\dot{a}}_{10}^{(m)} \big )^{2}+\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N (P_{i/m}^2 +Q_{i/m}^2 )\nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m}\right) } \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {[B_{i/m} P_{j/m} P_{l/m} \Delta _1 \left( i,j,l\right) } } } \nonumber \\&+\,B_{i/m} Q_{j/m} Q_{l/m} \Delta _2 \left( l,j,i\right) \nonumber \\&+\,2C_{i/m} P_{j/m} Q_{l/m} \Delta _2 \left( i,l,j\right) ], \end{aligned}$$

(64)

$$\begin{aligned} f_5^{(0)}= & {} a_{20}^{(m)} {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} \nonumber \\&+\,\frac{1}{2}\sum _{i=1}^N \left[ a_{20}^{(m)} \left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m}\right) \right. \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) \nonumber \\&\left. +\,{\dot{a}}_{20}^{(m)} \left( b_{2i/m} P_{i/m} +c_{2i/m} Q_{i/m} \right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} B_{j/m} P_{l/m} \Delta _1 \left( i,j,l\right) \right. \nonumber \\&+\,b_{2i/m} C_{j/m} Q_{l/m} \Delta _2 \left( l,j,i\right) \;\nonumber \\&+\,c_{2i/m} B_{j/m} Q_{l/m} \Delta _2 \left( i,l,j\right) \nonumber \\&\left. +\,c_{2i/m} C_{j/m} P_{l/m} \Delta _2 \left( i,j,l\right) \right] , \end{aligned}$$

(65)

$$\begin{aligned} f_6^{(0)}= & {} \big (a_{20}^{(m)}\big )^{2}{\dot{a}}_{20}^{(m)} +a_{20}^{(m)} \sum _{i=1}^N \left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N {(b_{2i/m}^2 +c_{2i/m}^2 )} \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} b_{2j/m} B_{l/m} \Delta _1 \left( i,j,l\right) \right. \nonumber \\&+\,2b_{2i/m} c_{2j/m} C_{k/m} \Delta _2 \left( l,j,i\right) \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} B_{l/m} \Delta _2 \left( i,j,l\right) \right] , \end{aligned}$$

(66)

$$\begin{aligned} f_7^{(0)}= & {} \big ({\dot{a}}_{10}^{(m)} \big )^{3}+\frac{3}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \big (P_{i/m}^2 +\,Q_{i/m}^2 \big )\nonumber \\&+\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3Q_{i/m} Q_{j/m} P_{l/m} \Delta _2 \left( i,j,l\right) \right. \nonumber \\&\left. +\,P_{i/m} P_{j/m} P_{l/m} \Delta _1 \left( i,j,l\right) \right] , \end{aligned}$$

(67)

$$\begin{aligned} f_8^{(0)}= & {} \big ({\dot{a}}_{10}^{(m)} \big )^{2}a_{20}^{(m)} +\frac{1}{2} \sum _{i=1}^N \big [a_{20}^{(m)} \big (P_{i/m}^2 +Q_{i/m}^2 \big ) \nonumber \\&+\,2{\dot{a}}_{10}^{(m)} \left( b_{(2)i/m} P_{i/m} +c_{(2)i/m} Q_{i/m} \right) \big ] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} P_{j/m} P_{l/m} \Delta _1 \left( i,j,l\right) \right. \nonumber \\&+\,b_{2i/m} Q_{j/m} Q_{l/m} \Delta _2 \left( l,j,i\right) \nonumber \\&\left. +\,2c_{(2)i/m} P_{j/m} Q_{l/m} \Delta _2 \left( i,l,j\right) \right] , \end{aligned}$$

(68)

$$\begin{aligned} f_9^{(0)}= & {} \big (a_{20}^{(m)}\big )^{2}{\dot{a}}_{10}^{(m)} +\,\frac{1}{2}\sum _{i=1}^N \big [2a_{20}^{(m)} \big (b_{2i/m} P_{i/m} \nonumber \\&+\,c_{2i/m} Q_{i/m} \big ) +\,{\dot{a}}_{10}^{(m)} \big (b_{2i/m}^2 +c_{2i/m}^2 \big )\big ] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 2b_{2i/m} c_{2j/m} Q_{l/m} \Delta _2 \left( l,j,i\right) \right. \nonumber \\&+\,b_{2i/m} b_{2j/m} P_{l/m} \Delta _1 \left( i,j,l\right) \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} P_{l/m} \Delta _2 \left( i,j,l\right) \right] , \end{aligned}$$

(69)

$$\begin{aligned} f_{10}^{(0)}= & {} \big (a_{20}^{(m)}\big )^{3}+\frac{3}{2}\sum _{i=1}^N {a_{20}^{(m)} (b_{2i/m}^2 +c_{2i/m}^2 )} \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3c_{2i/m} c_{2j/m} b_{2l/m} \Delta _2 \left( i,j,l\right) \right. \nonumber \\&\left. +\,b_{2i/m} b_{2j/m} b_{2l/m} \Delta _1 \left( i,j,l\right) \right] , \end{aligned}$$

(70)

$$\begin{aligned} f_{11}^{(0)}= & {} \big (a_{10}^{(m)} \big )^{3}+\frac{3}{2}\sum _{i=1}^N {a_{10}^{(m)} \big (b_{1i/m}^2 +c_{1i/m}^2 \big )} \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3c_{1i/m} c_{1j/m} b_{1l/m} \Delta _2 \left( i,j,l\right) \right. \nonumber \\&\left. +\,b_{1i/m} b_{1j/m} b_{1l/m} \Delta _1 \left( i,j,l\right) \right] . \end{aligned}$$

(71)

Define delta functions for cosine terms

$$\begin{aligned} \Delta _3 \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} +\delta _{i+l+k}^j +\delta _{i+j+l}^k +\delta _{k+j+l}^i +\delta _{k+i}^{l+j} \nonumber \\&+\,\delta _{i+j}^{l+k} +\delta _{k+i+j}^l , \nonumber \\ \Delta _4 \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} +\delta _{i+l+k}^j -\delta _{i+j+l}^k +\delta _{k+j+l}^i +\delta _{k+i}^{l+j}\nonumber \\&-\,\delta _{i+j}^{l+k} -\delta _{k+i+j}^l , \nonumber \\ \Delta _5 \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} +\delta _{i+l+k}^j -\delta _{i+j+l}^k -\delta _{k+j+l}^i \;-\;\delta _{k+i}^{l+j}\nonumber \\&+\,\delta _{i+j}^{l+k} +\delta _{k+i+j}^l , \nonumber \\ \Delta _6 \left( i,j,k,l\right)= & {} -\delta _{i+l}^{j+k} -\delta _{i+l+k}^j -\delta _{i+j+l}^k +\delta _{k+j+l}^i +\delta _{k+i}^{l+j}\nonumber \\&+\,\delta _{i+j}^{l+k} +\delta _{k+i+j}^l . \end{aligned}$$

(72)

The coefficients for cosine terms are

$$\begin{aligned} f_{1k}^{(c)}= & {} 3\big ({\dot{a}}_{20}^{(m)}\big )^{2}B_{k/m} \!+\! \frac{3}{2}{\dot{a}}_{10}^{(m)} \!\sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3C_{i/m} C_{j/m} B_{l/m} \Delta _4 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,B_{i/m} B_{j/m} B_{l/m} \Delta _3 \left( i,j,k,l\right) \right] , \end{aligned}$$

(73)

$$\begin{aligned} f_{2k}^{(c)}= & {} ({\dot{a}}_{20}^{(m)} )^{2}P_{k/m} +2{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} B_{k/m} \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ C_{i/m} C_{j/m} P_{l/m} \Delta _4 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,2B_{i/m} C_{j/m} Q_{l/m} \Delta _5 \left( i,j,k,l\right) \right] , \end{aligned}$$

(74)

$$\begin{aligned} f_{3k}^{(c)}= & {} 2{\dot{a}}_{20}^{(m)} a_{20}^{(m)} B_{k/m} +({\dot{a}}_{20}^{(m)} )^{2}b_{2k/m} \nonumber \\&+\,\frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} B_{j/m} B_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,b_{2i/m} C_{j/m} C_{l/m} \Delta _5 \left( i,j,k,l\right) \nonumber \\&\left. +\,2c_{2i/m} B_{j/m} C_{l/m} \Delta _6 \left( i,j,k,l\right) \right] , \end{aligned}$$

(75)

$$\begin{aligned} f_{4k}^{(c)}= & {} 2{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} P_{k/m} +\,\left( {\dot{a}}_{10}^{(m)} \right) ^{2}B_{k/m} \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ P_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,Q_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N [B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \nonumber \\&+\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) ] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ B_{i/m} P_{j/m} P_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,B_{i/m} Q_{j/m} Q_{l/m} \Delta _5 \left( i,j,k,l\right) \nonumber \\&\left. +\,C_{i/m} P_{j/m} Q_{l/m} \Delta _6 \left( i,j,k,l\right) \right] , \end{aligned}$$

(76)

$$\begin{aligned} f_{5k}^{(c)}= & {} a_{20}^{(m)} {\dot{a}}_{20}^{(m)} P_{k/m} +\,a_{20}^{(m)} {\dot{a}}_{10}^{(m)} B_{k/m} \nonumber \\&+\,\frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] +{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} b_{2k/m} \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} B_{j/m} P_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,b_{2i/m} C_{j/m} Q_{l/m} \Delta _5 \left( i,j,k,l\right) \nonumber \\&+\,c_{2i/m} B_{j/m} Q_{l/m} \Delta _6 \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} C_{j/m} P_{l/m} \Delta _4 \left( i,j,k,l\right) \right] , \end{aligned}$$

(77)

$$\begin{aligned} f_{6k}^{(c)}= & {} \big (a_{20}^{(m)}\big )^{2}B_{k/m} +2a_{20}^{(m)} {\dot{a}}_{20}^{(m)} b_{2k/m} \nonumber \\&+\,a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N\left[ b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} b_{2j/m} B_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,b_{2i/m} c_{2j/m} C_{l/m} \Delta _5 \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} B_{l/m} \Delta _4 \left( i,j,k,l\right) \right] , \end{aligned}$$

(78)

$$\begin{aligned} f_{7k}^{(c)}= & {} 3\big ({\dot{a}}_{10}^{(m)} \big )^2P_{k/m} {+}\frac{3}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N [P_{i/m} P_{j/m} \Delta _1 (i,j,k)\nonumber \\&+\,Q_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) ] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3Q_{i/m} Q_{j/m} P_{l/m} \Delta _4 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,P_{i/m} P_{j/m} P_{l/m} \Delta _3 \left( i,j,k,l\right) \right] , \end{aligned}$$

(79)

$$\begin{aligned} f_{8k}^{(c)}= & {} 2{\dot{a}}_{10}^{(m)} {\dot{a}}_{20}^{(m)} P_{k/m} \nonumber \\&+\,\frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ P_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,Q_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} P_{j/m} P_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,b_{2i/m} Q_{j/m} Q_{l/m} \Delta _5 \left( i,j,k,l\right) \nonumber \\&\left. +\,2c_{2i/m} P_{j/m} Q_{l/m} \Delta _6 \left( i,j,k,l\right) \right] , \end{aligned}$$

(80)

$$\begin{aligned} f_{9k}^{(c)}= & {} \big (a_{20}^{(m)}\big )^{2}P_{k/m} +2{\dot{a}}_{10}^{(m)} a_{20}^{(m)} b_{2k/m} \nonumber \\&+\,a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&+\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} \Delta _2 \left( i,j,k\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ b_{2i/m} b_{2j/m} P_{l/m} \Delta _3 \left( i,j,k,l\right) \right. \nonumber \\&+\,2b_{2i/m} c_{2j/m} Q_{l/m} \Delta _5 \left( i,j,k,l\right) +\nonumber \\&\left. +\,c_{2i/m} c_{2j/m} P_{l/m} \Delta _4 \left( i,j,k,l\right) \right] , \end{aligned}$$

(81)

$$\begin{aligned} f_{10}^{(c)}= & {} 3\big (a_{20}^{(m)}\big )^{2}b_{2k/m} +\frac{3}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N [b_{2i/m} b_{2j/m} \Delta _1 (i,j,k)\nonumber \\&+\,c_{2i/m} c_{2j/m} \Delta _3 (i,j,k)]\nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N\big [3c_{2i/m} c_{2j/m} b_{2l/m} \Delta _4 \left( i,j,k,l\right) \nonumber \\&+\,b_{2i/m} b_{2j/m} b_{2l/m} \Delta _3 \left( i,j,k,l\right) \big ], \end{aligned}$$

(82)

$$\begin{aligned} f_{11}^{(c)}= & {} 3\big (a_{10}^{(m)} \big )^{2}b_{1k/m} +\frac{3}{2}a_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \big [b_{1i/m} b_{1j/m} \Delta _1 (i,j,k)\nonumber \\&+\,c_{1i/m} c_{1j/m} \Delta _3 (i,j,k)\big ] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N 3c_{1i/m} c_{1j/m} b_{1l/m} \Delta _4 \left( i,j,k,l\right) \nonumber \\&+\,b_{1i/m} b_{1j/m} b_{1l/m} \Delta _3 \left( i,j,k,l\right) \big ]. \end{aligned}$$

(83)

Define delta functions for sine terms as follows

$$\begin{aligned} \Delta _7 \left( i,j,k,l\right)= & {} -\delta _{i+l}^{j+k} +\delta _{i+l+k}^j +\delta _{i+j+l}^k -\delta _{k+j+l}^i +\delta _{k+i}^{l+j} \nonumber \\&+\,\delta _{i+j}^{l+k} -\delta _{k+i+j}^l , \nonumber \\ \Delta _8 \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} -\delta _{i+l+k}^j -\delta _{i+j+l}^k -\delta _{k+j+l}^i +\delta _{k+i}^{l+j} \nonumber \\&+\,\delta _{i+j}^{l+k} -\delta _{k+i+j}^l , \nonumber \\ \Delta _9 \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} -\delta _{i+l+k}^j +\,\delta _{i+j+l}^k -\delta _{k+j+l}^i \nonumber \\&+\delta _{k+i}^{l+j} -\delta _{i+j}^{l+k} +\delta _{k+i+j}^l , \nonumber \\ \Delta _{10} \left( i,j,k,l\right)= & {} \delta _{i+l}^{j+k} -\delta _{i+l+k}^j +\,\delta _{i+j+l}^k +\delta _{k+j+l}^i \nonumber \\&-\delta _{k+i}^{l+j} +\delta _{i+j}^{l+k} -\delta _{k+i+j}^l. \end{aligned}$$

(84)

For since nonlinear terms, the coefficients functions are

$$\begin{aligned} f_{1k}^{(s)}= & {} 3({\dot{a}}_{20}^{(m)} )^{2}C_{k/m} +\,3{\dot{a}}_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} \Delta _2 \left( k,j,i\right) } } \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3B_{i/m} C_{j/m} B_{l/m} \Delta _7 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} C_{l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(85)

$$\begin{aligned} f_{2k}^{(s)}= & {} ({\dot{a}}_{20}^{(m)} )^{2}Q_{k/m} +2{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} C_{k/m} \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\,C_{i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} \Delta _2 \left( k,j,i\right) } }\nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ B_{i/m} B_{j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) \right. \nonumber \\&+\,C_{i/m} C_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) \nonumber \\&\left. +\,2B_{i/m} C_{j/m} P_{l/m} \Delta _7 \left( i,j,k,l\right) \right] , \end{aligned}$$

(86)

$$\begin{aligned} f_{3k}^{(s)}= & {} 2{\dot{a}}_{20}^{(m)} a_{20}^{(m)} C_{k/m} +({\dot{a}}_{20}^{(m)} )^{2}c_{2k/m} \nonumber \\&+\,a_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} } \Delta _2 \left( k,j,i\right) } \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) \right] \nonumber \\&+\,c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) ]\nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 2b_{2i/m} B_{j/m} C_{l/m} \Delta _9 \left( i,j,k,l\right) \right. \nonumber \\&+\,c_{2i/m} B_{j/m} B_{l/m} \Delta _{10} \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} C_{j/m} C_{l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(87)

$$\begin{aligned} f_{4k}^{(s)}= & {} 2{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} Q_{k/m} +\big ({\dot{a}}_{10}^{(m)} \big )^{2}C_{k/m} \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {P_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) } } \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&+\left. \,C_{i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 2B_{i/m} P_{j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) \right. \nonumber \\&+\,C_{i/m} P_{j/m} P_{l/m} \Delta _{10} \left( i,j,k,l\right) \nonumber \\&+\left. \,C_{i/m} Q_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(88)

$$\begin{aligned} f_{5k}^{(s)}= & {} a_{20}^{(m)} {\dot{a}}_{20}^{(m)} Q_{k/m} +a_{20}^{(m)} {\dot{a}}_{10}^{(m)} C_{k/m} +\,{\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} c_{2k/m} \nonumber \\&+\,\frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N [B_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \nonumber \\&+\,C_{i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2j/m} Q_{j/m} \Delta _2 \left( k,j,i\right) )\right. \nonumber \\&\left. +\,c_{2i/m} P_{j/m} \Delta _2 \left( i,k,j\right) )\right] \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) )\right. \nonumber \\&\left. +\,c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \big [b_{2i/m} B_{j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) \nonumber \\&+\,b_{2i/m} C_{j/m} P_{l/m} \Delta _7 \left( i,j,k,l\right) \nonumber \\&+\,c_{2i/m} B_{j/m} P_{l/m} \Delta _{10} \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} C_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) )\right] , \end{aligned}$$

(89)

$$\begin{aligned} f_{6k}^{(s)}= & {} \big (a_{20}^{(m)}\big )^{2}C_{k/m} +2a_{20}^{(m)} {\dot{a}}_{20}^{(m)} c_{2k/m} \nonumber \\&+\,a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) \right. \nonumber \\&\left. +\,b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) \right] \nonumber \\&+\,{\dot{a}}_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {c_{2i/m} b_{2j/m} \Delta _2 \left( i,k,j\right) } } \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N {\sum _{j=1}^N {\sum _{k=1}^N \left[ {b_{2i/m} b_{2j/m} C_{l/m} \Delta _9 \left( i,j,k,l\right) } \right. }} )\nonumber \\&+\,2b_{2i/m} c_{2j/m} B_{l/m} \Delta _7 \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} C_{l/m} \Delta _9 \left( i,j,k,l\right) )\right] , \end{aligned}$$

(90)

$$\begin{aligned} f_{7k}^{(s)}= & {} 3\big ({\dot{a}}_{10}^{(m)} \big )^{2}Q_{k/m} \nonumber \\&+\,3{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {P_{j/m} Q_{k/m} } \Delta _2 \left( k,j,i\right) } \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3P_{i/m} Q_{j/m} P_{l/m} \Delta _7 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,Q_{i/m} Q_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(91)

$$\begin{aligned} f_{8k}^{(s)}= & {} 2{\dot{a}}_{10}^{(m)} a_{20}^{(m)} Q_{k/m} +a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N P_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\big ({\dot{a}}_{10}^{(m)} \big )^{2}c_{2k/m} +{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} Q_{j/m} \Delta _2 (k,j,i) )\right. \nonumber \\&\left. +\,c_{2i/m} P_{j/m} \Delta _2 (i,k,j))\right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N \left[ 2b_{2i/m} P_{j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) )\right. } }\nonumber \\&+\,c_{2i/m} P_{j/m} P_{l/m} \Delta _{10} \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) )\right] , \end{aligned}$$

(92)

$$\begin{aligned} f_{9k}^{(s)}= & {} \big (a_{20}^{(m)}\big )^{2}Q_{k/m} +2{\dot{a}}_{10}^{(m)} a_{20}^{(m)} c_{2k/m} \nonumber \\&+\,{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {b_{2i/m} c_{2j/m} \Delta _2 \left( k,j,i\right) } } \nonumber \\&+\,a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\,c_{2i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 2b_{2i/m} c_{2j/m} P_{l/m} \Delta _7 \left( i,j,k,l\right) \right. \nonumber \\&+\,b_{2i/m} b_{2j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(93)

$$\begin{aligned} f_{10k}^{(s)}= & {} 3\big (a_{20}^{(m)}\big )^{2}c_{2k/m} \nonumber \\&+3a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N {b_{2i/m} c_{2j/m} } \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N \sum _{j=1}^N \sum _{l=1}^N \left[ 3b_{2i/m} c_{2j/m} b_{2l/m} \Delta _7 \left( i,j,k,l\right) \right. \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} c_{2l/m} \Delta _8 \left( i,j,k,l\right) \right] , \end{aligned}$$

(94)

$$\begin{aligned} f_{11k}^{(s)}= & {} 3\big (a_{10}^{(m)} \big )^{2}c_{1k/m} \nonumber \\&+3a_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N {b_{1i/m} c_{1j/m} } \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\frac{1}{4}\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {\left[ 3b_{1i/m} c_{1j/m} b_{1l/m} \Delta _7 \left( i,j,k,l\right) \right. } } } \nonumber \\&\left. +\,c_{1i/m} c_{1j/m} c_{1l/m} \Delta _8 \left( i,j,k,l\right) \right] . \end{aligned}$$

(95)

Derivatives of coefficients with displacement

Derivatives of $f_\lambda ^{(0)}$ ($\lambda =1,2,\ldots ,11)$ with respect to $z_r$ will be given. The first term for the constant coefficient is

$$\begin{aligned} g_{1r}^{(0)} =g_{1r}^{(0)} (1)+\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {\frac{1}{4}g_{1r}^{(0)} \left( i,j,l,q\right) } } }} \end{aligned}$$

(96)

where

$$\begin{aligned} g_{1r}^{(0)} (1)= & {} 3{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) [\delta _{i+3N+1}^r B_{i/m} \nonumber \\&-\,\delta _{i+2N+1}^r C_{i/m} \left( \frac{i\Omega }{m}\right) ] , \nonumber \\ g_{1r}^{(0)} \left( i,j,l,1\right)= & {} 3\left[ -\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} B_{l/m} \right. \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} B_{l/m} \nonumber \\&\left. +\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) C_{i/m} C_{j/m} \right] \Delta _2 \left( i,j,l\right) ,\nonumber \\ g_{1r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) B_{j/m} B_{l/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) B_{i/m} B_{j/m} ]\Delta _1 \left( i,j,l\right) .\nonumber \\ \end{aligned}$$

(97)

The second term for the constant coefficient is

$$\begin{aligned} g_{2r}^{(0)}= & {} g_{2r}^{(0)} (1)+g_{2r}^{(0)} (2)\nonumber \\&+\,\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{2r}^{(0)} \left( i,j,l,q\right) } } } } \end{aligned}$$

(98)

where

$$\begin{aligned} g_{2r}^{(0)} (1)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) \left[ \delta _{i+3N+1}^r P_{i/m} +\delta _{i+N}^r B_{i/m} \right. \nonumber \\&\left. -\,\delta _{i+2N+1}^r Q_{i/m} -\delta _i^r C_{i/m} \right] , \nonumber \\&g_{2r}^{(0)} (2)={\dot{a}}_{10}^{(m)} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) \left[ \delta _{i+3N+1}^r B_{i/m} \right. \nonumber \\&\left. -\delta _{l+2N+1}^r C_{i/m} \right] , \nonumber \\ g_{2r}^{(0)} \left( i,j,l,1\right)= & {} \left[ \left( \frac{i\Omega }{m}\right) \delta _{i+3N+1}^r B_{j/m} P_{l/m}\right. \nonumber \\&+\,\left( \frac{j\Omega }{m}\right) \delta _{j+3N+1}^r B_{i/m} P_{l/m} \nonumber \\&\left. +\,\left( \frac{l\Omega }{m}\right) \delta _{l+N}^r B_{i/m} B_{j/m} \right] \Delta _1 \left( i,j,l\right) , \nonumber \\ g_{2r}^{(0)} \left( i,j,l,2\right)= & {} \left[ -\left( \frac{i\Omega }{m}\right) \delta _{i+2N+1}^r C_{j/m} P_{l/m} \right. \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _{j+2N+1}^r C_{i/m} P_{l/m} \nonumber \\&\left. +\,\left( \frac{l\Omega }{m}\right) \delta _{l+N}^r C_{i/m} C_{j/m} \right] \Delta _2 \left( i,j,l\right) , \nonumber \\ g_{2r}^{(0)} \left( i,j,l,3\right)= & {} 2\left[ \left( \frac{i\Omega }{m}\right) \delta _{i+3N+1}^r C_{j/m} Q_{l/m}\right. \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _{j+2N+1}^r B_{i/m} Q_{l/m} \nonumber \\&\left. -\,\left( \frac{l\Omega }{m}\right) \delta _l^r B_{i/m} C_{j/m} \right] \Delta _2 \left( l,j,i\right) \}. \end{aligned}$$

(99)

The third term for the constant coefficient is

$$\begin{aligned} g_{3r}^{(0)} =\sum _{p=1}^3 {g_{3r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{3r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(100)

where

$$\begin{aligned} g_{3r}^{(0)} (1)= & {} a_{20}^{(m)} ({\dot{a}}_{20}^{(m)} )^{2}, \nonumber \\ g_{3r}^{(0)} (2)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) \left( \delta _{i+3N+1}^r B_{i/m} \right. \nonumber \\&\left. -\,\delta _{i+2N+1}^r C_{i/m} \right) , \nonumber \\ g_{3r}^{(0)} (3)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left[ (\delta _{i+2N+1}^r B_{i/m} +\delta _{i+3N+1}^r C_{i/m} )\right. \nonumber \\&+\,\left. \left( \frac{i\Omega }{m}\right) (\delta _{i+3N+1}^r b_{2i/m} -\delta _{i+2N+1}^r c_{2i/m} )\right] \nonumber \\ g_{3r}^{(0)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r B_{j/m} B_{l/m} \right. \nonumber \\&+\,\left( \frac{j\Omega }{m}\right) \delta _{j+3N+1}^r b_{2i/m} B_{l/m} \nonumber \\&+\,\left. \left( \frac{l\Omega }{m}\right) \delta _{l+3N+1}^r b_{2i/m} B_{j/m} \right] \Delta _2 \left( i,j,l\right) , \nonumber \\ g_{3r}^{(0)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r C_{j/m} C_{l/m} \right. \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _{j+2N+1}^r b_{2i/m} C_{l/m}\nonumber \\&-\,\left. \left( \frac{l\Omega }{m}\right) \delta _{l+2N+1}^r b_{2i/m} C_{j/m} \right] \Delta _2 \left( l,j,i\right) , \nonumber \\ g_{3r}^{(0)} \left( i,j,l,3\right)= & {} 2\left[ \delta _{i+3N+1}^r B_{j/m} C_{l/m} \right. \nonumber \\&+\,\left( \frac{j\Omega }{m}\right) \delta _{j+3N+1}^r c_{2i/m} C_{l/m} \nonumber \\&-\,\left. \left( \frac{l\Omega }{m}\right) \delta _{l+2N+1}^r c_{2i/m} B_{j/m} \right] \Delta _2 \left( i,l,j\right) .\nonumber \\ \end{aligned}$$

(101)

The fourth term for the constant coefficient is

$$\begin{aligned} g_{4r}^{(0)} =\sum _{p=1}^2 {g_{4r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{4r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(102)

where

$$\begin{aligned} g_{4r}^{(0)} (1)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) [\delta _{i+N}^r P_{i/m} -\delta _i^r Q_{i/m} ), \nonumber \\ g_{4r}^{(0)} (2)= & {} {\dot{a}}_{10} \sum _{i=1}^N \left( \frac{i\Omega }{m}\right) (\delta _{i+3N+1}^r P_{i/m} \nonumber \\&+\,\delta _{i+N}^r B_{i/m} -\delta _{i+2N+1}^r Q_{i/m} -\delta _i^r C_{i/m}), \nonumber \\ g_{4r}^{(0)} \left( i,j,l,1\right)= & {} \left[ \left( \frac{i\Omega }{m}\right) \delta _{i+3N+1}^r P_{j/m} P_{l/m} \right. \nonumber \\&+\,\left( \frac{j\Omega }{m}\right) \delta _{j+N}^r B_{i/m} P_{l/m} \nonumber \\&+\,\left. \left( \frac{l\Omega }{m}\right) \delta _{l+N}^r B_{i/m} P_{j/m} \right] \Delta _1 \left( i,j,l\right) , \nonumber \\ g_{4r}^{(0)} \left( i,j,l,2\right)= & {} \left[ \left( \frac{i\Omega }{m}\right) \delta _{i+3N+1}^r Q_{j/m} Q_{l/m}\right. \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _j^r B_{i/m} Q_{l/m}\nonumber \\&-\,\left. \left( \frac{l\Omega }{m}\right) \delta _l^r B_{i/m} Q_{j/m} \right] \Delta _2 \left( l,j,i\right) , \nonumber \\ g_{4r}^{(0)} \left( i,j,l,3\right)= & {} -2\left[ \left( \frac{i\Omega }{m}\right) \delta _{i+2N+1}^r P_{j/m} Q_{l/m} \right. \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _{j+N}^r C_{i/m} Q_{l/m} \nonumber \\&+\,\left. \left( \frac{l\Omega }{m}\right) \delta _l^r C_{i/m} P_{j/m} \right] \Delta _2 \left( i,l,j\right) ].\nonumber \\ \end{aligned}$$

(103)

The fifth term for the constant coefficient is

$$\begin{aligned} g_{5r}^{(0)} =\sum _{p=1}^4 {g_{5r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^4 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{5r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(104)

where

$$\begin{aligned} g_{5r}^{(0)} (1)= & {} \delta _{2N+1}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ g_{5r}^{(0)} (2)= & {} \frac{1}{2}\sum _{i=1}^N {\{\delta _{2N+1}^r \left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m}\right) } \nonumber \\&+\,a_{20}^{(m)} \left( \frac{i\Omega }{m}\right) [\delta _{i+3N+1}^r P_{i/m} +\delta _{i+N}^r B_{i/m} ]\nonumber \\&-\,a_{20}^{(m)} \left( \frac{i\Omega }{m}\right) [\delta _{i+2N+1}^r Q_{i/m} +C_{i/m} \delta _i^r Q_{i/m} ]\}, \nonumber \\ g_{5r}^{(0)} (3)= & {} {\dot{a}}_{10} \{\left[ \delta _{i+2N+1}^r B_{i/m} +\left( \frac{i\Omega }{m}\right) \delta _{i+3N+1}^r b_{2i/m} \right] \nonumber \\&+\,[\delta _{i+3N+1}^r C_{i/m} -\left( \frac{i\Omega }{m}\right) \delta _{i+2N+1}^r c_{2i/m} ]\}, \nonumber \\ g_{5r}^{(0)} (4)= & {} {\dot{a}}_{20} \{[\delta _{i+2N+1}^r P_{i/m} +\left( \frac{i\Omega }{m}\right) \delta _{i+N}^r b_{2i/m} ]\nonumber \\&+\,[\delta _{i+3N+1}^r Q_{i/m} -\left( \frac{i\Omega }{m}\right) \delta _i^r c_{(2)i/m} )]\}, \nonumber \\ g_{5r}^{(0)} \left( i,j,l,1\right)= & {} [\delta _{i+2N+1}^r B_{j/m} P_{l/m} +\left( \frac{j\Omega }{m}\right) \delta _{j+3N+1}^r b_{2i/m} P_{l/m} \nonumber \\&+\,\left( \frac{l\Omega }{m}\right) \delta _{l+N}^r b_{(2)i/m} B_{j/m} ]\Delta _1 \left( i,j,l\right) , \nonumber \\ g_{5r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+2N+1}^r C_{j/m} Q_{l/m} -\left( \frac{j\Omega }{m}\right) \delta _{j+2N+1}^r b_{2i/m} Q_{l/m} \nonumber \\&-\,\left( \frac{l\Omega }{m}\right) \delta _l^r b_{2i/m} C_{j/m} ]\Delta _2 \left( l,j,i\right) , \nonumber \\ g_{5r}^{(0)} \left( i,j,l,3\right)= & {} [\delta _{i+3N+1}^r B_{j/m} Q_{l/m} +\left( \frac{j\Omega }{m}\right) \delta _{j+3N+1}^r c_{2i/m} Q_{l/m} \nonumber \\&-\,\left( \frac{l\Omega }{m}\right) \delta _l^r c_{2i/m} B_{j/m} ]\Delta _2 \left( i,l,j\right) ,\nonumber \\ g_{5r}^{(0)} \left( i,j,l,4\right)= & {} [\delta _{i+3N+1}^r C_{j/m} P_{l/m} \nonumber \\&-\,\left( \frac{j\Omega }{m}\right) \delta _{j+2N+1}^r c_{2i/m} P_{l/m} \nonumber \\&+\,\left( \frac{l\Omega }{m}\right) \delta _{l+N}^r c_{2i/m} C_{j/m} ]\Delta _2 \left( i,j,l\right) . \end{aligned}$$

(105)

The sixth term for the constant coefficient is

$$\begin{aligned} g_{6r}^{(0)} =\sum _{p=1}^4 {g_{6r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{6r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(106)

where

$$\begin{aligned} g_{6r}^{(0)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ g_{6r}^{(0)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) } , \nonumber \\ g_{6r}^{(0)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \{[\delta _{i+2N+1}^r B_{i/m} \nonumber \\&+ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) b_{2i/m} ]\nonumber \\&+\,[\delta _{i+3N+1}^r c_{2i/m} C_{i/m} -\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) c_{2i/m} ]\}, \nonumber \\ g_{6r}^{(0)} (4)= & {} {\dot{a}}_{20} \sum _{i=1}^N (\delta _{i+2N+1}^r b_{2i/m} \nonumber \\&+\,\delta _{i+3N+1}^r c_{2i/m} ) , \nonumber \\ g_{6r}^{(0)} \left( i,j,l,1\right)= & {} [\delta _{i+2N+1}^r b_{2j/m} B_{l/m} +\delta _{j+2N+1}^r b_{2i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} ]\Delta _1 \left( i,j,l\right) , \nonumber \\ g_{6r}^{(0)} \left( i,j,l,2\right)= & {} 2[\delta _{i+2N+1}^r c_{2j/m} C_{k/m}\nonumber \\&+\,\delta _{j+3N+1}^r b_{2i/m} C_{k/m} \nonumber \\&-\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} ]\Delta _2 \left( l,j,i\right) , \nonumber \\ g_{6r}^{(0)} \left( i,j,l,3\right)= & {} [\delta _{i+3N+1}^r c_{2j/m} B_{l/m} +\delta _{j+3N+1}^r c_{2i/m} B_{l/m}\nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) c_{(2)i/m} c_{(2)j/m} ]\Delta _2 \left( i,j,l\right) ].\nonumber \\ \end{aligned}$$

(107)

The seventh term for the constant coefficient is

$$\begin{aligned} g_{7r}^{(0)} =g_{7r}^{(0)} (1)+\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{7r}^{(0)} \left( i,j,l,q\right) } } } } \end{aligned}$$

(108)

where

$$\begin{aligned} g_{7r}^{(0)} (1)= & {} 3{\dot{a}}_{10} \sum _{i=1}^N {\left( \frac{i\Omega }{m}\right) (\delta _{i+N}^r P_{i/m} -\delta _i^r Q_{i/m} )} , \nonumber \\ g_{7r}^{(0)} \left( i,j,l,1\right)= & {} 3[-\delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m} P_{l/m} \nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) Q_{i/m} Q_{j/m} P_{l/m} ]\Delta _2 \left( i,j,l\right) , \nonumber \\ g_{7r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) P_{j/m} P_{k/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) P_{i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) P_{i/m} P_{j/m} ]\Delta _1 \left( i,j,l\right) .\nonumber \\ \end{aligned}$$

(109)

The eighth term for the constant coefficient is

$$\begin{aligned} g_{8r}^{(0)} =\sum _{p=1}^2 {g_{8r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{8r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(110)

where

$$\begin{aligned} g_{8r}^{(0)} (1)= & {} \big ({\dot{a}}_{10}^{(m)} \big )^{2}\delta _{2N+1}^r , \nonumber \\ g_{8r}^{(0)} (2)= & {} \frac{1}{2}\sum _{i=1}^N \{[\delta _{2N+1}^r (P_{i/m}^2 +Q_{i/m}^2 ) \nonumber \\&+\,2a_{20}^{(m)} \left( \frac{i\Omega }{m}\right) (\delta _{i+N}^r P_{i/m} +\delta _i^r Q_{i/m} )] \nonumber \\&+\,2{\dot{a}}_{10} [\delta _{i+2N+1}^r P_{i/m} +\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) b_{2i/m}\nonumber \\&+\,\delta _{i+3N+1}^r Q_{i/m} -\delta _i^r \left( \frac{i\Omega }{m}\right) c_{2i/m} ]\}, \nonumber \\ g_{8r}^{(0)} \left( i,j,l,1\right)= & {} \delta _{i+2N+1}^r P_{j/m} P_{l/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} P_{j/m} ]\Delta _1 \left( i,j,l\right) , \nonumber \\ g_{8r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+2N+1}^r Q_{j/m} Q_{l/m} \nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) b_{2i/m} Q_{l/m}\nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} Q_{j/m} ]\Delta _2 \left( l,j,i\right) , \nonumber \\ g_{8r}^{(0)} \left( i,j,l,3\right)= & {} 2\left[ \delta _{i+3N+1}^r P_{j/m} Q_{l/m} \right. \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} P_{j/m} Q_{l/m}\nonumber \\&-\left. \,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} P_{j/m} \right] \Delta _2 \left( i,l,j\right) ].\nonumber \\ \end{aligned}$$

(111)

The ninth term for the constant coefficient is

$$\begin{aligned} g_{9r}^{(0)} =\sum _{p=1}^2 {g_{9r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{9r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(112)

where

$$\begin{aligned} g_{9r}^{(0)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ g_{9r}^{(0)} (2)= & {} \frac{1}{2}\sum _{i=1}^N {[2\delta _{2N+1}^r \left( b_{2i/m} P_{i/m} +c_{2i/m} Q_{i/m} \right) } ]\nonumber \\&+\,2a_{20}^{(m)} [\delta _{i+2N+1}^r P_{i/m} +\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) b_{2i/m} \nonumber \\&+\,\delta _{i+3N+1}^r Q_{i/m} -\delta _i^r \left( \frac{i\Omega }{m}\right) c_{2i/m} ]\nonumber \\&+\,{\dot{a}}_{10} [\delta _{i+2N+1}^r b_{2i/m} +\delta _{i+3N+1}^r c_{2i/m} ], \nonumber \\ g_{9r}^{(0)} \left( i,j,l,1\right)= & {} 2[\delta _{i+2N+1}^r c_{2j/m} Q_{l/m} +\delta _{j+3N+1}^r b_{2i/m} Q_{l/m} \nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} Q_{l/m} ]\Delta _2 \left( l,j,i\right) , \nonumber \\ g_{9r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+2N+1}^r b_{2j/m} P_{l/m} +\delta _{j+2N+1}^r b_{2i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} ]\Delta _1 \left( i,j,l\right) ,\nonumber \\ g_{9r}^{(0)} \left( i,j,l,3\right)= & {} [\delta _{i+3N+1}^r c_{2j/m} P_{l/m} +\delta _{i+3N+1}^r c_{2i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} c_{2j/m} ]\Delta _2 \left( i,j,l\right) . \end{aligned}$$

(113)

The tenth term for the constant coefficient is

$$\begin{aligned} g_{10r}^{(0)} =\sum _{p=1}^2 {g_{10r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{10r}^{(0)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(114)

where

$$\begin{aligned} g_{10r}^{(0)} (1)= & {} 3\left( a_{20}^{(m)}\right) ^{2}\delta _{2N+1}^r , \nonumber \\ g_{10r}^{(0)} (2)= & {} \frac{3}{2}\sum _{i=1}^N {[\delta _{2N+1}^r (b_{2i/m}^2 +c_{2i/m}^2 )} \nonumber \\&+\,2a_{20}^{(m)} (\delta _{i+2N+1}^r b_{2i/m} +\delta _{i+3N+1}^r c_{2i/m} )],\nonumber \\ g_{10r}^{(0)} \left( i,j,l,1\right)= & {} 3[\delta _{i+3N+1}^r c_{2j/m} b_{2l/m} +\delta _{j+3N+1}^r c_{2i/m} b_{2l/m} \nonumber \\&+\,\delta _{l+2N+1}^r c_{2i/m} c_{2j/m} ]\Delta _2 \left( i,j,l\right) ,\nonumber \\ g_{10r}^{(0)} \left( i,j,l,2\right)= & {} [\delta _{i+2N+1}^r b_{2j/m} b_{2l/m} +\delta _{j+2N+1}^r b_{2i/m} b_{2l/m} \nonumber \\&+\,\delta _{l+2N+1}^r b_{2i/m} b_{2j/m} ]\Delta _1 \left( i,j,l\right) .\nonumber \\ \end{aligned}$$

(115)

The eleventh term for the constant coefficient of the transverse motion is

$$\begin{aligned} g_{11r}^{(0)} =\sum _{p=1}^2 {g_{11r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{11r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(116)

where

$$\begin{aligned} g_{11r}^{(0)} (1)= & {} 3\big (a_{10}^{(m)} \big )^{2}\delta _0^r , \\ g_{11r}^{(0)} (2)= & {} \frac{3}{2}\sum _{i=1}^N {\big [\delta _0^r (b_{1i/m}^2 +c_{1i/m}^2 )} +2a_{10}^{(m)} (\delta _i^r b_{1i/m} \\&+\,\delta _{i+N}^r c_{1i/m} )\big ], \end{aligned}$$

$$\begin{aligned} g_{11r}^{(0)} \left( i,j,l,1\right)= & {} 3\big [\delta _{i+N}^r c_{1j/m} b_{1l/m} +\delta _{j+N}^r c_{1i/m} b_{1l/m}\nonumber \\&+\,\delta _l^r c_{1i/m} c_{1j/m} \big ]\Delta _2 \left( i,j,l\right) , \nonumber \\ g_{11r}^{(0)} \left( i,j,l,2\right)= & {} \big [\delta _i^r b_{1j/m} b_{1l/m} +\delta _j^r b_{1i/m} b_{1l/m} \nonumber \\&+\,\delta _l^r b_{1i/m} b_{1j/m} \big ]\Delta _1 \left( i,j,l\right) . \end{aligned}$$

(117)

For cosine term derivatives, the first term for the cosine coefficient is

$$\begin{aligned} g_{1kr}^{(c)} =\sum _{p=1}^2 {g_{1kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{1kr}^{(c)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(118)

where

$$\begin{aligned} g_{1kr}^{(c)} (1)= & {} 3\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) ({\dot{a}}_{20}^{(m)} )^{2}, \nonumber \\ g_{1kr}^{(c)} (2)= & {} \frac{3}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) B_{j/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} \nonumber \\&+\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{1kr}^{(c)} \left( i,j,l,1\right)= & {} 3[-\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} B_{l/m}\nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) C_{i/m} C_{j/m} ]\Delta _4 \left( i,j,k,l\right) , \nonumber \\ g_{1kr}^{(c)} \left( i,j,l,2\right)= & {} [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) B_{j/m} B_{l/m}\nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) B_{i/m} B_{j/m} ]\Delta _3 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(119)

The second term for the cosine coefficient is

$$\begin{aligned} g_{2kr}^{(c)} =\sum _{p=1}^2 {g_{2kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{2kr}^{(c)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(120)

where

$$\begin{aligned} g_{2kr}^{(c)} (1)= & {} \delta _{k+N}^r \left( \frac{k\Omega }{m}\right) ({\dot{a}}_{20}^{(m)} )^{2}\nonumber \\&+\,2\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ g_{2kr}^{(c)} (2)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) B_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&-\,\left[ \delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \right. \nonumber \\&\left. +\delta _j^r \left( \frac{j\Omega }{m}\right) C_{i/m} \right] \Delta _2 \left( i,j,k\right) \} \nonumber \\&+\,\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) B_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} \right] \Delta _1 \left( i,j,k\right) \; \nonumber \\&-\,\left[ \delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{2kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ -\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} P_{l/m}\right. \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} P_{l/m} \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) C_{i/m} C_{j/m} \right] \Delta _4 \left( i,j,k,l\right) , \nonumber \\ g_{2kr}^{(c)} \left( i,j,l,2\right)= & {} 2\left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} Q_{l/m} \right. \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} Q_{l/m} \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) B_{i/m} C_{j/m} \right] \Delta _5 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(121)

The third term for the cosine coefficient is

$$\begin{aligned} g_{3kr}^{(c)} =\sum _{p=1}^4 {g_{3kr}^{(c)} (p)} {+}\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{3kr}^{(c)} (i,j,l,q)}}}} \end{aligned}$$

(122)

where

$$\begin{aligned} g_{3kr}^{(c)} (1)= & {} 2\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{20}^{(m)} a_{20}^{(m)} \nonumber \\&+\,2\delta _{k+2N+1}^r {\dot{a}}_{20}^{(m)} B_{k/m} \nonumber \\&+\,\delta _{k+2N+1}^r ({\dot{a}}_{20}^{(m)} )^{2},\nonumber \\ g_{3kr}^{(c)} (2)= & {} \frac{1}{2}\delta _{k+2N+1}^r \sum _{i=1}^N \sum _{j=1}^N [B_{i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \nonumber \\&+\,C_{i/m} C_{j/m} \Delta _2 \left( i,j,k\right) ] , \nonumber \\ g_{3kr}^{(c)} (3)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) B_{j/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} \nonumber \\&+\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{3kr}^{(c)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+2N+1}^r B_{j/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&+\,[\delta _{i+3N+1}^r C_{j/m} \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} ]\Delta _2 \left( i,j,k\right) \},\nonumber \end{aligned}$$

$$\begin{aligned} g_{3kr}^{(c)} \left( i,j,l,1\right)= & {} [\delta _{i+2N+1}^r B_{j/m} B_{l/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} B_{j/m} ]\Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{3kr}^{(c)} \left( i,j,l,2\right)= & {} \;[\delta _{i+2N+1}^r C_{j/m} C_{l/m} \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} C_{l/m} \nonumber \\&-\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} C_{j/m} ]\Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{3kr}^{(c)} \left( i,j,l,3\right)= & {} 2[\delta _{i+3N+1}^r B_{j/m} C_{l/m} \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} C_{l/m} \nonumber \\&-\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} B_{j/m} ]\Delta _6 \left( i,j,k,l\right) \}.\nonumber \\ \end{aligned}$$

(123)

The fourth term for the cosine coefficient is

$$\begin{aligned} g_{4kr}^{(c)} =\sum _{p=1}^3 {g_{4kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 \sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{4kr}^{(c)} \left( i,j,l,q\right) }}}\nonumber \\ \end{aligned}$$

(124)

where

$$\begin{aligned} g_{4kr}^{(c)} (1)= & {} 2\delta _{k+N}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} \nonumber \\&+\,\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) \left( {\dot{a}}_{10}^{(m)} \right) ^{2}, \nonumber \\ g_{4kr}^{(c)} (2)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) P_{j/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) P_{i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m}\nonumber \\&+\,\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{4kr}^{(c)} (3)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) B_{i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \nonumber \\&+\,\delta _j^r \left( \frac{j\Omega }{m}\right) C_{i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{4kr}^{(c)} \left( i,j,l,1\right)= & {} [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} P_{l/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) B_{i/m} P_{l/m} \nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) B_{i/m} P_{j/m} ]\Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{4kr}^{(c)} \left( i,j,l,2\right)= & {} [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} Q_{l/m}\nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) B_{i/m} Q_{l/m} \nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) B_{i/m} Q_{j/m} ]\Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{4kr}^{(c)} \left( i,j,l,3\right)= & {} [-\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} Q_{l/m} \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) C_{i/m} Q_{l/m} \nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) C_{i/m} P_{j/m} ]\Delta _6 \left( i,j,k,l\right) . \end{aligned}$$

(125)

The fifth term for the cosine coefficient is

$$\begin{aligned} g_{5kr}^{(c)} =\sum _{p=1}^4 {g_{5kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{5kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(126)

where

$$\begin{aligned} g_{5kr}^{(c)} (1)= & {} \delta _{2N+1}^r {\dot{a}}_{20}^{(m)} P_{k/m}\nonumber \\&+\,\delta _{k+N}^r \left( \frac{k\Omega }{m}\right) a_{20}^{(m)} {\dot{a}}_{20}^{(m)} +\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} B_{k/m}\nonumber \\&+\,\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) a_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ g_{5kr}^{(c)} (2)= & {} \frac{1}{2}\delta _{k+N}^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ g_{5kr}^{(c)} (3)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m}\nonumber \right. \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) B_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&-\,\left[ \delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{i/m} \right. \nonumber \\&\left. +\,\delta _j^r \left( \frac{j\Omega }{m}\right) C_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{5kr}^{(c)} (4)= & {} \delta _{k+2N+1}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)}\nonumber \\&+\,\frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \bigg \{\left[ \delta _{i+2N+1}^r P_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r Q_{j/m} -\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,j,k\right) \bigg \}, \nonumber \\ g_{5kr}^{(c)} (5)= & {} \;\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \bigg \{\left[ \delta _{i+2N+1}^r B_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r C_{j/m} -\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,j,k\right) \bigg \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{5kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r B_{j/m} P_{l/m} +\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} B_{j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(c)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r C_{j/m} Q_{l/m} -\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} Q_{l/m}\right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} C_{j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(c)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r B_{j/m} Q_{l/m} +\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} Q_{l/m} \right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} B_{j/m} \right] \Delta _6 \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(c)} \left( i,j,l,4\right)= & {} \left[ \delta _{i+3N+1}^r C_{j/m} P_{l/m} -\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} C_{j/m} \right] \Delta _4 \left( i,j,k,l\right) \}. \end{aligned}$$

(127)

The sixth term for the cosine coefficient is

$$\begin{aligned} g_{6kr}^{(c)} =\sum _{p=1}^4 {g_{6kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 \sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{6kr}^{(c)} \left( i,j,l,q\right) }}}\nonumber \\ \end{aligned}$$

(128)

where

$$\begin{aligned} g_{6kr}^{(c)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} B_{k/m} +\delta _{k+3N+1}^r \left( \frac{k\Omega }{m}\right) \left( a_{20}^{(m)}\right) ^{2}\nonumber \\&+\,2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} b_{2k/m}+2\delta _{k+2N+1}^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ g_{6kr}^{(c)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N [b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \nonumber \\&+\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) ] , \nonumber \\ g_{6kr}^{(c)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+2N+1}^r B_{j/m}\nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&+\,[\delta _{i+3N+1}^r Q_{j/m} \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{6kr}^{(c)} (4)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+2N+1}^r b_{2j/m} \nonumber \\&+\,\delta _{j+2N+1}^r b_{2i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&+\,[\delta _{i+3N+1}^r c_{2j/m} +\delta _{j+3N+1}^r c_{2i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{6kr}^{(c)} \left( i,j,l,1\right)= & {} [\delta _{i+2N+1}^r b_{2j/m} B_{l/m} +\delta _{j+2N+1}^r b_{2i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} ]\Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{6kr}^{(c)} \left( i,j,l,2\right)= & {} [\delta _{i+2N+1}^r c_{2j/m} C_{l/m} +\delta _{j+3N+1}^r b_{2i/m} C_{l/m} \nonumber \\&-\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} ]\Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{6kr}^{(c)} \left( i,j,l,3\right)= & {} [\delta _{i+3N+1}^r c_{2j/m} B_{l/m} +\delta _{j+3N+1}^r c_{2i/m} B_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} c_{2j/m} ]\Delta _4 \left( i,j,k,l\right) ].\nonumber \\ \end{aligned}$$

(129)

The seventh term for the cosine coefficient is

$$\begin{aligned} g_{7kr}^{(c)} =\sum _{p=1}^2 {g_{7kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{7kr}^{(c)} \left( i,j,l,q\right) }}}}\nonumber \\ \end{aligned}$$

(130)

where

$$\begin{aligned} g_{7kr}^{(c)} (1)= & {} 3\delta _{k+N}^r \left( \frac{k\Omega }{m}\right) \left( {\dot{a}}_{10}^{(m)} \right) { }^2 \nonumber \\ g_{7kr}^{(c)} (2)= & {} \frac{3}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \left[ \delta _{i+N}^r \left( \frac{i\Omega }{m}\right) P_{j/m}\right. \right. \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) P_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m}\nonumber \\&\left. +\,\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{j/m} ]\Delta _2 \left( i,j,k\right) \right\} , \nonumber \end{aligned}$$

$$\begin{aligned} g_{7kr}^{(c)} \left( i,j,l,1\right)= & {} 3\left[ -\delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m} P_{l/m} -\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{i/m} P_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) Q_{i/m} Q_{j/m} \right] \Delta _4 \left( i,j,k,l\right) , \nonumber \\ g_{7kr}^{(c)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+N}^r \left( \frac{i\Omega }{m}\right) P_{j/m} P_{l/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) P_{i/m} P_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) P_{i/m} P_{j/m} \right] \Delta _3 \left( i,j,k,l\right) . \end{aligned}$$

(131)

The eighth term for the cosine coefficient is

$$\begin{aligned} g_{8kr}^{(c)} =\sum _{p=1}^3 {g_{8kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{8kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(132)

where

$$\begin{aligned} g_{8kr}^{(c)} (1)= & {} 2\delta _{k+N}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{10}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ g_{8kr}^{(c)} (2)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) P_{j/m}\nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) P_{i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&-\,[\delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \nonumber \\&+\,\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{i/m} ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{8kr}^{(c)} (3)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r P_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _1 \left( i,j,k\right) \;\; \nonumber \\&+\,\left[ \delta _{i+3N+1}^r Q_{j/m} \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{8kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r P_{j/m} P_{l/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} P_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r (\frac{l\Omega }{m})b_{2i/m} P_{j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{8kr}^{(c)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r Q_{j/m} Q_{l/m} -\delta _j^r \left( \frac{j\Omega }{m}\right) b_{2i/m} Q_{l/m}\right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} Q_{j/m} Q_{l/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{8kr}^{(c)} \left( i,j,l,3\right)= & {} 2\left[ \delta _{i+3N+1}^r P_{j/m} Q_{l/m}\right. \nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} Q_{l/m}\nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} P_{j/m} \right] \Delta _6 \left( i,j,k,l\right) ]. \end{aligned}$$

(133)

The ninth term for the cosine coefficient is

$$\begin{aligned} g_{9kr}^{(c)} =\sum _{p=1}^4 {g_{9kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{9kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(134)

where

$$\begin{aligned} g_{9kr}^{(c)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} P_{k/m} +\delta _{k+N}^r \big (\frac{k\Omega }{m}\big )\big (a_{20}^{(m)}\big )^{2}\nonumber \\&+\,2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} b_{2k/m} +2\delta _{k+2N+1}^r {\dot{a}}_{10}^{(m)} a_{20}^{(m)} , \nonumber \\ g_{9kr}^{(c)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&+\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) , \nonumber \\ g_{9kr}^{(c)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r P_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r Q_{j/m} -\delta _j^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ g_{9kr}^{(c)} (4)= & {} \frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r b_{2j/m} \right. \nonumber \\&\left. +\,\delta _{j+2N+1}^r b_{2j/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r c_{2j/m} +\delta _{j+3N+1}^r c_{2i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{9kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r b_{2j/m} P_{l/m} +\delta _{j+2N+1}^r b_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ g_{9kr}^{(c)} \left( i,j,l,2\right)= & {} 2\left[ \delta _{i+2N+1}^r c_{2j/m} Q_{l/m} +\delta _{j+3N+1}^r b_{2i/m} Q_{l/m}\right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ g_{9kr}^{(c)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r c_{2j/m} P_{l/m} +\delta _{j+3N+1}^r c_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} c_{2j/m} \right] \Delta _4 \left( i,j,k,l\right) \}.\nonumber \\ \end{aligned}$$

(135)

The tenth term for the cosine coefficient is

$$\begin{aligned} g_{10kr}^{(c)} =\sum _{p=1}^3 {g_{10kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{10kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(136)

where

$$\begin{aligned} g_{10kr}^{(c)} (1)= & {} 6\delta _{2N+1}^r a_{20}^{(m)} b_{2k/m} +3\delta _{k+2N+1}^r \big (a_{20}^{(m)} \big )^{2}, \nonumber \\ g_{10kr}^{(c)} (2)= & {} \frac{3}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} \Delta _3 \left( i,j,k\right) \right] , \nonumber \\ g_{10kr}^{(c)} (3)= & {} \frac{3}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r b_{2j/m} \right. \nonumber \\&\left. +\,\delta _{j+2N+1}^r b_{2i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r c_{2j/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r c_{2i/m} \right] \Delta _3 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{10kr}^{(c)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+3N+1}^r c_{2j/m} b_{2l/m}\right. \nonumber \\&+\delta _{j+3N+1}^r c_{2i/m} b_{2l/m}\nonumber \\&\left. +\,\delta _{l+2N+1}^r c_{2i/m} c_{2j/m} \right] \Delta _4 \left( i,j,k,l\right) , \nonumber \\ g_{10kr}^{(c)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r b_{2j/m} b_{2l/m} \right. \nonumber \\&\left. +\,\delta _{j+2N+1}^r b_{2i/m} b_{2l/m}\right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r b_{2i/m} b_{2j/m} \right] \Delta _3 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(137)

The eleventh term for the cosine coefficient is

$$\begin{aligned} g_{11kr}^{(c)} =\sum _{p=1}^3 {g_{11kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{11kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(138)

where

$$\begin{aligned} g_{11kr}^{(c)} (1)= & {} 6\delta _0^r a_{10}^{(m)} b_{1k/m} +3\delta _k^r \big (a_{10}^{(m)} \big )^{2}, \nonumber \\ g_{11kr}^{(c)} (2)= & {} \frac{3}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N [b_{1i/m} b_{1j/m} \Delta _1 \left( i,j,k\right) \nonumber \\&+\,c_{1i/m} c_{1j/m} \Delta _3 \left( i,j,k\right) ] , \nonumber \\ g_{11kr}^{(c)} (3)= & {} \frac{3}{2}a_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{[\delta _{i+N}^r b_{1j/m} \nonumber \\&+\,\delta _{j+N}^r b_{1i/m} ]\Delta _1 \left( i,j,k\right) \nonumber \\&+\,[\delta _{i+N}^r c_{1j/m} +\,\delta _{j+N}^r c_{1i/m} ]\Delta _3 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} g_{11kr}^{(c)} \left( i,j,l,1\right)= & {} 3[\delta _{i+N}^r c_{1j/m} b_{1l/m} +\,\delta _{j+N}^r c_{1i/m} b_{1l/m} \nonumber \\&+\,\delta _{l+N}^r c_{1i/m} c_{1j/m} ]\Delta _4 \left( i,j,k,l\right) , \nonumber \\ g_{11kr}^{(c)} \left( i,j,l,2\right)= & {} [\delta _{i+N}^r b_{1j/m} b_{1l/m} +\,\delta _{j+N}^r b_{1i/m} b_{1l/m}\nonumber \\&+\,\delta _{l+N}^r b_{1i/m} b_{1j/m} ]\Delta _3 \left( i,j,k,l\right) . \end{aligned}$$

(139)

For derivatives for nonlinear sine terms, the first term for the sine coefficient is

$$\begin{aligned} g_{1kr}^{(s)} =\sum _{p=1}^2 {g_{1kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{1kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(140)

where

$$\begin{aligned} g_{1kr}^{(s)} (1)= & {} -3\delta _{k+2N+1}^r \left( \frac{k\Omega }{m}\right) \big ({\dot{a}}_{20}^{(m)} \big )^{2}, \nonumber \\ g_{1kr}^{(s)} (2)= & {} 3{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m}\right. \nonumber \\&\left. -\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} \right] \Delta _2 \left( k,j,i\right) ,\nonumber \\ g_{1kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} B_{l/m} \right. \nonumber \\&\left. -\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} B_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) B_{i/m} C_{j/m} \right] \Delta _7 \left( i,j,k,l\right) ,\nonumber \\ \nonumber \\ g_{1kr}^{(s)} \left( i,j,l,2\right)= & {} -\left[ \delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m} C_{l/m}\right. \nonumber \\&+\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) C_{i/m} C_{l/m} \nonumber \\&\left. +\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) C_{i/m} C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(141)

The second term for the sine coefficient is

$$\begin{aligned} g_{2kr}^{(s)} =\sum _{p=1}^3 {g_{2kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{2kr}^{(s)} \left( i,j,l,1\right) } } } \end{aligned}$$

(142)

where

$$\begin{aligned} g_{2kr}^{(s)} (1)= & {} -\delta _k^r \left( \frac{k\Omega }{m}\right) \big ({\dot{a}}_{20}^{(m)} \big )^{2}\nonumber \\&-\,2\delta _{k+2N+1}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} \nonumber \\ g_{2kr}^{(s)} (2)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\big [B_{i/m} Q_{j/m} \nonumber \\&+B_{i/m} Q_{j/m} \big ]\Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ C_{i/m} P_{j/m} +C_{i/m} P_{j/m} \right] \Delta _2 \left( i,k,j\right) \} \nonumber \\ g_{2kr}^{(s)} (3)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} \Delta _2 \left( k,j,i\right) } } \; \nonumber \end{aligned}$$

$$\begin{aligned} g_{2kr}^{(s)} \left( i,j,l,1\right)= & {} B_{i/m} B_{j/m} Q_{l/m} \Delta _9 \left( i,j,k,l\right) \nonumber \\&+\,C_{i/m} C_{j/m} Q_{l/m} \Delta _8 \left( i,j,k,l\right) \nonumber \\&+\,2B_{i/m} C_{j/m} P_{l/m} \Delta _7 \left( i,j,k,l\right) \end{aligned}$$

(143)

The third term for the sine coefficient is

$$\begin{aligned} g_{3kr}^{(s)} =\sum _{p=1}^4 {g_{3kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{3kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(144)

where

$$\begin{aligned} g_{3kr}^{(s)} (1)= & {} 2{\dot{a}}_{20}^{(m)} \delta _{2N+1}^r C_{k/m} \nonumber \\&-\,2\delta _{k+2N+1}^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{20}^{(m)} a_{20}^{(m)} \nonumber \\&+\,\big ({\dot{a}}_{20}^{(m)} \big )^{2}\delta _{k+3N+1}^r , \nonumber \\ g_{3kr}^{(s)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} } \Delta _2 \left( k,j,i\right) } , \nonumber \\ g_{3kr}^{(s)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) C_{j/m}\right. \nonumber \\&\left. -\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) B_{i/m} \right] \Delta _2 \left( k,j,i\right) , \nonumber \\ g_{3kr}^{(s)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Bigg \{\left[ \delta _{i+2N+1}^r C_{j/m} \right. \nonumber \\&\left. -\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{j+3N+1}^r B_{j/m} \right. \nonumber \\&\left. \left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,k,j\right) \right\} , \nonumber \end{aligned}$$

$$\begin{aligned} g_{3kr}^{(s)} \left( i,j,l,1\right)= & {} 2\left[ \delta _{i+2N+1}^r B_{j/m} C_{l/m} \right. \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} C_{l/m} \nonumber \\&\left. -\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} B_{j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{3kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{j+3N+1}^r B_{j/m} B_{l/m}\right. \nonumber \\&+\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} B_{l/m} \nonumber \\&\left. +\,\delta _{l+3N+1}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} B_{j/m} \right] \Delta _{10} \left( i,j,k,l\right) , \nonumber \\ g_{3kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{j+3N+1}^r C_{j/m} C_{l/m}\right. \nonumber \\&-\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} C_{l/m} \nonumber \\&\left. -\,\delta _{l+2N+1}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(145)

The fourth term for the sine coefficient is

$$\begin{aligned} g_{4kr}^{(s)} =\sum _{p=1}^3 {g_{4kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{4kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(146)

where

$$\begin{aligned} g_{4kr}^{(s)} (1)= & {} -2\delta _k^r \Big (\frac{k\Omega }{m}\Big ){\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)}\nonumber \\&-\,\delta _{k+2N+1}^r \Big (\frac{k\Omega }{m}\Big )\left( {\dot{a}}_{10}^{(m)} \right) ^{2} \nonumber \\ g_{4kr}^{(s)} (2)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big [\delta _{i+N}^r \left( \frac{i\Omega }{m}\right) Q_{j/m}\nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) P_{i/m} \Big ]\Delta _2 \left( k,j,i\right) , \nonumber \\ g_{4kr}^{(s)} (3)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \Big [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \right. \nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) B_{i/m} \Big ]\Delta _2 \left( k,j,i\right) \nonumber \\&+\,\Big [-\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) C_{i/m} \Big ]\Delta _2 \left( i,k,j\right) \right\} , \nonumber \end{aligned}$$

$$\begin{aligned} g_{4kr}^{(s)} \left( i,j,l,1\right)= & {} 2\Big [\delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} Q_{l/m}\nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) B_{i/m} Q_{l/m}\nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) B_{i/m} P_{j/m} \Big ]\Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{4kr}^{(s)} \left( i,j,l,2\right)= & {} \Big [-\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} P_{l/m}\nonumber \\&+\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) C_{i/m} P_{l/m}\nonumber \\&+\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) C_{i/m} P_{j/m} \Big ]\Delta _{10} \left( i,j,k,l\right) , \nonumber \\ g_{4kr}^{(s)} \left( i,j,l,3\right)= & {} \Big [-\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} Q_{l/m} \nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) C_{i/m} Q_{l/m} \nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m}\right) C_{i/m} Q_{j/m} \Big ]\Delta _8 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(147)

The fifth term for the sine coefficient is

$$\begin{aligned} g_{5kr}^{(s)} =\sum _{p=1}^6 {g_{5kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{5kr}^{(s)} \left( i,j,l,q\right) } } } } \end{aligned}$$

(148)

where

$$\begin{aligned} g_{5kr}^{(s)} (1)= & {} \delta _{2N+1}^r {\dot{a}}_{20}^{(m)} Q_{k/m} \nonumber \\&-\,\delta _k^r \left( \frac{k\Omega }{m}\right) a_{20}^{(m)} {\dot{a}}_{20}^{(m)} +\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} C_{k/m} \nonumber \\&-\,\delta _{k+2N+1}^r \left( \frac{k\Omega }{m}\right) a_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ g_{5kr}^{(s)} (2)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} Q_{j/m} +B_{i/m} Q_{j/m} \right] \nonumber \\&\Delta _2 \left( k,j,i\right) , \nonumber \\ g_{5kr}^{(s)} (3)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N} \left[ \delta _{i+3N+1}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) C_{i/m} \right] \nonumber \\&\times \, \Delta _2 \left( i,k,j\right) +\left[ -\delta _{i+2N+1}^r \left( \frac{i\Omega }{m}\right) P_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) C_{i/m} \right] \Delta _2 \left( i,k,j\right) \}, \nonumber \\ g_{5kr}^{(s)} (4)= & {} {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} \delta _{i+3N+1}^r , \nonumber \\ g_{5kr}^{(s)} (5)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r Q_{j/m} \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) b_{2j/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r P_{j/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,k,j\right) \}, \nonumber \\ g_{5kr}^{(s)} (6)= & {} \frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r C_{j/m} \right. \nonumber \\&\left. -\,\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r B_{j/m} +\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,k,j\right) \}, \nonumber \\ g_{5kr}^{(s)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r B_{j/m} Q_{l/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} Q_{l/m}\right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} B_{j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r C_{j/m} P_{l/m} -\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} C_{j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r B_{j/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} B_{j/m} \right] \Delta _{10} \left( i,j,k,l\right) , \nonumber \\ g_{5kr}^{(s)} \left( i,j,l,4\right)= & {} \left[ \delta _{i+3N+1}^r C_{j/m} Q_{l/m}-\delta _{j+2N+1}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} Q_{l/m} \right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(149)

The sixth term for the sine coefficient is

$$\begin{aligned} g_{6kr}^{(s)} =\sum _{p=1}^4 {g_{6kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{6kr}^{(s)} \left( i,j,l,q\right) } } } } \end{aligned}$$

(150)

where

$$\begin{aligned} g_{6kr}^{(s)} (1)= & {} 2\delta _{2N+1} a_{20}^{(m)} C_{k/m} -\delta _{k+2N+1} \left( \frac{k\Omega }{m}\right) \left( a_{20}^{(m)} \right) ^{2} \nonumber \\&+\,2\delta _{2N+1} {\dot{a}}_{20}^{(m)} c_{2k/m} +2\delta _{k+3N+1} a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ g_{6kr}^{(s)} (2)= & {} \delta _{2N+1} \sum _{i=1}^N \sum _{j=1}^N \left[ c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) \right. \nonumber \\&\left. +\,b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) \right] , \nonumber \\ g_{6kr}^{(s)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+3N+1} B_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1} \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,k,j\right) \nonumber \\&+\,\left[ \delta _{i+2N+1} C_{j/m} -\delta _{j+2N+1} \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _2 \left( k,j,i\right) \}, \nonumber \\ g_{6kr}^{(s)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+3N+1} b_{2j/m} +\delta _{j+2N+1} c_{2i/m} \right] \Delta _2 \left( i,k,j\right) ,\; \nonumber \end{aligned}$$

$$\begin{aligned} g_{6kr}^{(s)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1} b_{2j/m} C_{l/m} +\delta _{j+2N+1} b_{2i/m} C_{l/m} \right. \nonumber \\&\left. -\,\delta _{l+2N+1} \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{6kr}^{(s)} \left( i,j,l,2\right)= & {} 2\left[ \delta _{i+2N+1} c_{2j/m} B_{l/m} +\delta _{j+3N+1} b_{2i/m} B_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+3N+1} \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{6kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1} c_{2j/m} C_{l/m} +\delta _{j+3N+1} c_{2i/m} C_{l/m} \right. \nonumber \\&\left. -\,\delta _{l+2N+1} \left( \frac{l\Omega }{m}\right) c_{2i/m} c_{2j/m} \right] \Delta _9 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(151)

The seventh term for the sine coefficient is

$$\begin{aligned} g_{7kr}^{(s)} =\sum _{p=1}^2 {g_{7kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{7kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(152)

where

$$\begin{aligned} g_{7kr}^{(s)} (1)= & {} -3\delta _k^r \left( \frac{k\Omega }{m}\right) \left( {\dot{a}}_{10}^{(m)} \right) ^{2}, \nonumber \\ g_{7kr}^{(s)} (2)= & {} 3{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+N}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) P_{i/m} \right] \Delta _2 \left( k,j,i\right) ,\nonumber \\ g_{7kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+N}^r \left( \frac{i\Omega }{m}\right) Q_{j/m} P_{l/m}\right. \nonumber \\&-\,\delta _j^r \left( \frac{j\Omega }{m}\right) P_{i/m} P_{l/m}\nonumber \\&-\,\delta _l^r \left( \frac{l\Omega }{m})P_{i/m} Q_{j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{7kr}^{(s)} \left( i,j,l,2\right)= & {} -\left[ \delta _i^r \left( \frac{i\Omega }{m}\right) Q_{j/m} Q_{l/m} +\delta _j^r \left( \frac{j\Omega }{m}\right) Q_{i/m} Q_{l/m} \right. \nonumber \\&\left. +\,\delta _l^r \left( \frac{l\Omega }{m}\right) Q_{i/m} Q_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(153)

The eighth term for the sine coefficient is

$$\begin{aligned} g_{8kr}^{(s)} =\sum _{p=1}^5 {g_{8kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{8kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(154)

where

$$\begin{aligned} g_{8kr}^{(s)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} Q_{k/m} -2\delta _k^r \left( \frac{k\Omega }{m}\right) {\dot{a}}_{10}^{(m)} a_{20}^{(m)} , \nonumber \\ g_{8kr}^{(s)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {P_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) } } , \nonumber \\ g_{8kr}^{(s)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+N}^r \left( \frac{i\Omega }{m}\right) Q_{j/m}\right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) P_{i/m} \right] \Delta _2 \left( k,j,i\right) , \nonumber \\ g_{8kr}^{(s)} (4)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+2N+1}^r Q_{j/m} \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) b_{2j/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{j+3N+1}^r P_{j/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m}\right] \Delta _2 \left( i,k,j\right) , \nonumber \\ g_{8kr}^{(s)} (5)= & {} \big ({\dot{a}}_{10}^{(m)} \big )^{2}\delta _{k+2N+1}^r , \nonumber \end{aligned}$$

$$\begin{aligned} g_{8kr}^{(s)} \left( i,j,l,1\right)= & {} 2\left[ \delta _{j+2N+1}^r P_{j/m} Q_{l/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) b_{2i/m} Q_{l/m} \right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} P_{j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{8kr}^{(s)} \left( i,j,l,2\right)= & {} \;\left[ \delta _{i+3N+1}^r P_{j/m} P_{l/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} P_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) c_{2i/m} P_{j/m} \right] \Delta _{10} \left( i,j,k,l\right) , \nonumber \\ g_{8kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r Q_{j/m} Q_{l/m} -\delta _j^r \left( \frac{j\Omega }{m}\right) c_{2i/m} Q_{l/m}\right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} Q_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(155)

The ninth term for the sine coefficient is

$$\begin{aligned} g_{9kr}^{(s)} =\sum _{p=1}^4 {g_{9kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{9kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(156)

where

$$\begin{aligned} g_{9kr}^{(s)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} Q_{k/m} -\delta _k^r \left( \frac{k\Omega }{m}\right) \left( a_{20}^{(m)}\right) ^{2}\nonumber \\&+\,2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} c_{2k/m}+\,2\delta _{k+3N+1}^r {\dot{a}}_{10}^{(m)} a_{20}^{(m)} , \nonumber \\ g_{9kr}^{(s)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\,c_{2i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ g_{9kr}^{(s)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r Q_{j/m} \right. \nonumber \\&\left. -\,\delta _j^r \left( \frac{j\Omega }{m}\right) b_{2i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r P_{j/m} +\delta _{j+N}^r \left( \frac{j\Omega }{m}\right) c_{2i/m} \right] \Delta _2 \left( i,k,j\right) \}, \nonumber \\ g_{9kr}^{(s)} (4)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+2N+1}^r c_{2j/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r b_{2i/m} \right] \Delta _2 \left( k,j,i\right) , \nonumber \\ g_{9kr}^{(s)} \left( i,j,l,1\right)= & {} 2\left[ \delta _{i+2N+1}^r c_{2j/m} P_{l/m} +\delta _{j+3N+1}^r b_{2i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r \left( \frac{l\Omega }{m}\right) b_{2i/m} c_{2j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{9kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r b_{2j/m} Q_{l/m} +\delta _{j+2N+1}^r b_{2i/m} Q_{l/m} \right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) b_{2i/m} b_{2j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ g_{9kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r c_{2j/m} Q_{l/m} +\delta _{j+3N+1}^r c_{2i/m} Q_{l/m} \right. \nonumber \\&\left. -\,\delta _l^r \left( \frac{l\Omega }{m}\right) c_{2i/m} c_{2j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(157)

The tenth term for the sine coefficient is

$$\begin{aligned} g_{10kr}^{(s)} =\sum _{p=1}^3 {g_{10kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{10kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(158)

where

$$\begin{aligned} g_{10kr}^{(s)} (1)= & {} 6\delta _{2N+1}^r a_{20}^{(m)} c_{2k/m} \nonumber \\&+\,3\delta _{k+3N+1}^r \big (a_{20}^{(m)}\big )^{2}, \nonumber \\ g_{10kr}^{(s)} (2)= & {} 3\delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {b_{2i/m} c_{2j/m} } \Delta _2 \left( k,j,i\right) } , \nonumber \\ g_{10kr}^{(s)} (3)= & {} 3a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+2N+1}^r c_{2j/m} \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r b_{2i/m} \right] \Delta _2 \left( k,j,i\right) , \nonumber \end{aligned}$$

$$\begin{aligned} g_{10kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+2N+1}^r c_{2j/m} b_{2l/m} \right. \nonumber \\&+\,\delta _{j+3N+1}^r b_{2i/m} b_{2l/m} \nonumber \\&\left. +\,\delta _{l+2N+1}^r b_{2i/m} c_{2j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{10kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+3N+1}^r c_{2j/m} c_{2l/m}\right. \nonumber \\&+\delta _{j+3N+1}^r c_{2i/m} c_{2l/m} \nonumber \\&\left. +\,\delta _{l+3N+1}^r c_{2i/m} c_{2j/m} \right] \Delta _8 (i,j,k,l). \end{aligned}$$

(159)

The eleventh term for the sine coefficient is

$$\begin{aligned} g_{11kr}^{(s)} =\sum _{p=1}^3 {g_{11kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {g_{11kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(160)

where

$$\begin{aligned} g_{11kr}^{(s)} (1)= & {} 6\delta _0^r a_{10}^{(m)} c_{1k/m}+\,3\delta _{k+N}^r \left( a_{10}^{(m)} \right) ^{2}, \nonumber \\ g_{11kr}^{(s)} (2)= & {} 3a_{10}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {b_{1i/m} c_{1j/m} } \Delta _2 \left( k,j,i\right) } , \nonumber \\ g_{11kr}^{(s)} (3)= & {} 3\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _i^r b_{1i/m} c_{1j/m} \right. \nonumber \\&\left. +\,\delta _{i+N}^r b_{1i/m} c_{1j/m} \right] \Delta _2 \left( k,j,i\right) , \nonumber \\ g_{11kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _i^r c_{1j/m} b_{1l/m} +\delta _{j+N}^r b_{1i/m} b_{1l/m} \right. \nonumber \\&\left. +\,\delta _l^r b_{1i/m} c_{1j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ g_{11kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+N}^r c_{1j/m} c_{1l/m} +\delta _{j+N}^r c_{1i/m} c_{1l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r c_{1i/m} c_{1j/m} \right] \Delta _8 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(161)

1.1 Derivatives of coefficients with velocity

For derivatives of $f_\lambda ^{(0)}$ ($\lambda =1,2,\ldots ,11)$ with ${\dot{z}}_r$, the first term of the constant is

$$\begin{aligned} h_{1r}^{(0)} =\sum _{p=1}^3 {h_{1r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{1r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(162)

where

$$\begin{aligned} h_{1r}^{(0)} (1)= & {} 3\delta _{2N+1}^r \big ({\dot{a}}_{20}^{(m)} \big )^{2} \nonumber \\ h_{1r}^{(0)} (2)= & {} \frac{3}{2}\delta _{2N+1}^r \sum _{i=1}^N {\big (B_{i/m}^2 +C_{i/m}^2 \big )} \nonumber \\ h_{1r}^{(0)} (3)= & {} 3{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \delta _{i+2N+1}^r B_{i/m}+\,\delta _{i+3N+1}^r C_{i/m} \right) \nonumber \\ h_{1r}^{(0)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+3N+1}^r C_{j/m} B_{l/m} +\delta _{j+3N+1}^r C_{i/m} B_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r C_{i/m} C_{j/m} \right] \Delta _2 \left( i,j,l\right) , \nonumber \\ h_{1r}^{(0)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r B_{j/m} B_{l/m} +\delta _{j+2N+1}^r B_{i/m} B_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r B_{i/m} B_{j/m} \right] \Delta _1 \left( i,j,l\right) . \end{aligned}$$

(163)

The second term of the constant is

$$\begin{aligned} h_{2r}^{(0)} =\sum _{p=1}^5 {h_{2r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{2r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(164)

where

$$\begin{aligned} h_{2r}^{(0)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} +\delta _0^r \big ({\dot{a}}_{20}^{(m)}\big )^{2}, \nonumber \\ h_{2r}^{(0)} (2)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m}\right) } , \nonumber \\ h_{2r}^{(0)} (3)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \left( \delta _{i+2N+1}^r P_{i/m} +\delta _i^r B_{i/m} \right. \nonumber \\&\left. +\,\delta _{i+3N+1}^r Q_{i/m} +\delta _{i+N}^r C_{i/m} \right) , \nonumber \\ h_{2r}^{(0)} (4)= & {} \frac{1}{2}\delta _0^r \sum _{i=1}^N {(B_{i/m}^2 +C_{i/m}^2 )} , \nonumber \\ h_{2r}^{(0)} (5)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \left( \delta _{i+2N+1}^r B_{i/m}+\,\delta _{i+3N+1}^r C_{i/m} \right) , \nonumber \end{aligned}$$

$$\begin{aligned} h_{2r}^{(0)} \left( i,j,l,1\right)= & {} \Big [\delta _{i+2N+1}^r B_{j/m} P_{l/m}+\,\delta _{j+2N+1}^r B_{i/m} P_{l/m}\nonumber \\&+\,\delta _l^r B_{i/m} B_{j/m} \Big ]\Delta _1 \left( i,j,l\right) , \nonumber \\ h_{2r}^{(0)} \left( i,j,l,2\right)= & {} \;\Big [\delta _{i+3N+1}^r C_{j/m} P_{l/m} +\delta _{j+3N+1}^r C_{i/m} P_{l/m} \nonumber \\&+\,\delta _l^r C_{i/m} C_{j/m} \Big ]\Delta _2 \left( i,j,l\right) , \nonumber \\ h_{2r}^{(0)} \left( i,j,l,3\right)= & {} 2\Big [\delta _{i+2N+1}^r C_{j/m} Q_{l/m}+\,\delta _{j+3N+1}^r B_{i/m} Q_{l/m} \nonumber \\&+\delta _{l+N}^r B_{i/m} C_{j/m} \Big ]\Delta _2 \left( l,j,i\right) \}. \end{aligned}$$

(165)

The third term of the constant is

$$\begin{aligned} h_{3r}^{(0)} =\sum _{p=1}^4 {h_{3r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{3r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(166)

where

$$\begin{aligned} h_{3r}^{(0)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ h_{3r}^{(0)} (2)= & {} a_{10}^{(m)} \sum _{i=1}^N {(\delta _{i+2N+1}^r B_{i/m} +} \delta _{i+3N+1}^r C_{i/m} ), \nonumber \\ h_{3r}^{(0)} (3)= & {} \delta _0^r \sum _{i=1}^N {\left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) } , \nonumber \\ h_{3r}^{(0)} (4)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \left( \delta _{i+2N+1}^r b_{2i/m} \right. \nonumber \\&\left. +\,\delta _{i+3N+1}^r c_{2i/m} \right) , \nonumber \\ h_{3r}^{(0)} \left( i,j,l,1\right)= & {} \Big [\delta _{j+2N+1}^r b_{2i/m} B_{l/m} \nonumber \\&+\,\delta _{l+2N+1}^r b_{2i/m} B_{j/m} \Big ]\Delta _2 \left( i,j,l\right) , \nonumber \\ h_{3r}^{(0)} \left( i,j,l,2\right)= & {} \Big [\delta _{j+3N+1}^r b_{2i/m} C_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r b_{2i/m} C_{j/m} \Big ]\Delta _2 \left( l,j,i\right) , \nonumber \\ h_{3r}^{(0)} \left( i,j,l,3\right)= & {} 2\Big [\delta _{j+2N+1}^r c_{2i/m} C_{l/m} \nonumber \\&+\,\delta _{l+3N+1}^r c_{2i/m} B_{j/m} \Big ]\Delta _2 \left( i,l,j\right) . \end{aligned}$$

(167)

The fourth term of the constant is

$$\begin{aligned} h_{4r}^{(0)} =\sum _{p=1}^4 {h_{4r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^4 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{4r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(168)

where

$$\begin{aligned} h_{4r}^{(0)} (1)= & {} \delta _{2N+1}^r \big ({\dot{a}}_{10}^{(m)} \big )^{2}+2\delta _0^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ h_{4r}^{(0)} (2)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N {(P_{i/m}^2 +Q_{i/m}^2 )} , \nonumber \\ h_{4r}^{(0)} (3)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N {(\delta _i^r P_{i/m} +\delta _{i+N}^r Q_{i/m} )} , \nonumber \\ h_{4r}^{(0)} (4)= & {} \delta _0^r \sum _{i=1}^N {\left( B_{i/m} P_{i/m} +C_{i/m} Q_{i/m}\right) } , \nonumber \\ h_{4r}^{(0)} (5)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \left[ \delta _{i+2N+1}^r P_{i/m} +\delta _i^r B_{i/m} \right. \nonumber \\&+\,\delta _{i+3N+1}^r Q_{i/m} +\delta _{i+N}^r C_{i/m}) , \nonumber \\ h_{4r}^{(0)} \left( i,j,l,1\right)= & {} \Big [\delta _{i+2N+1}^r P_{j/m} P_{l/m}+\,\delta _j^r B_{i/m} P_{l/m} \nonumber \\&+\,\delta _l^r B_{i/m} P_{j/m} \Big ]\Delta _1 \left( i,j,l\right) , \nonumber \\ h_{4r}^{(0)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r Q_{j/m} Q_{l/m} +\delta _{j+N}^r B_{i/m} Q_{l/m} \right. \nonumber \\&\left. +\delta _{l+N}^r B_{i/m} Q_{j/m} \right] \Delta _2 \left( l,j,i\right) , \nonumber \\ h_{4r}^{(0)} \left( i,j,l,3\right)= & {} 2\left[ \delta _{i+3N+1}^r P_{j/m} Q_{l/m} +\delta _j^r C_{i/m} Q_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r C_{i/m} P_{j/m} \right] \Delta _2 \left( i,l,j\right) . \end{aligned}$$

(169)

The fifth term of the constant is

$$\begin{aligned} h_{5r}^{(0)} =\sum _{p=1}^5 {h_{5r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{5r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(170)

where

$$\begin{aligned} h_{5r}^{(0)} (1)= & {} \delta _{2N+1}^r a_{20}^{(m)} {\dot{a}}_{10}^{(m)} \nonumber \\&+\,\delta _0^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ h_{5r}^{(0)} (2)= & {} \frac{1}{2}\sum _{i=1}^N a_{20}^{(m)} \left( \delta _{i+2N+1}^r P_{i/m} +\delta _i^r B_{i/m} \right. \nonumber \\&\left. +\,\delta _{i+3N+1}^r Q_{i/m} +\delta _{i+N}^r C_{i/m} \right) , \nonumber \\ h_{5r}^{(0)} (3)= & {} \frac{1}{2}\sum _{i=1}^N \left[ \delta _0^r \left( b_{2i/m} B_{i/m} +c_{2i/m} C_{i/m} \right) \right. \nonumber \\&\left. +\,{\dot{a}}_{10}^{(m)} \left( \delta _{i+2N+1}^r b_{2i/m} +\delta _{i+3N+1}^r c_{2i/m} \right) \right] , \nonumber \\ h_{5r}^{(0)} (4)= & {} \frac{1}{2}\sum _{i=1}^N \left[ \delta _{2N+1}^r \left( b_{2i/m} P_{i/m} +c_{2i/m} Q_{i/m} \right) \right. \nonumber \\&\left. +\,{\dot{a}}_{20}^{(m)} (\delta _i^r b_{2i/m} +\delta _{i+N}^r c_{2i/m} )\right] , \nonumber \\ h_{5r}^{(0)} \left( i,j,l,1\right)= & {} b_{2i/m} \left[ \delta _{j+2N+1}^r P_{l/m} +\delta _l^r B_{j/m} \right] \Delta _1 \left( i,j,l\right) , \nonumber \\ h_{5r}^{(0)} \left( i,j,l,2\right)= & {} b_{2i/m} \left[ \delta _{j+3N+1}^r Q_{l/m} +\delta _{l+N}^r C_{j/m} \right] \Delta _2 \left( l,j,i\right) ,\nonumber \\ h_{5r}^{(0)} \left( i,j,l,3\right)= & {} c_{2i/m} \left[ \delta _{j+2N+1}^r Q_{l/m} +\delta _{l+N}^r B_{j/m} \right] \Delta _2 \left( i,l,j\right) , \nonumber \\ h_{5r}^{(0)} \left( i,j,l,4\right)= & {} c_{2i/m} \left[ \delta _{j+3N+1}^r P_{l/m} +\delta _l^r C_{j/m} \right] \Delta _2 \left( i,j,l\right) .\nonumber \\ \end{aligned}$$

(171)

The sixth term of the constant is

$$\begin{aligned} h_{6r}^{(0)} =\sum _{p=1}^3 {h_{6r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{6r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(172)

where

$$\begin{aligned} h_{6r}^{(0)} (1)= & {} \delta _{2N+1}^r \Big (a_{20}^{(m)}\Big )^{2}, \nonumber \\ h_{6r}^{(0)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N {\left( \delta _{i+2N+1}^r b_{2i/m} +\delta _{i+3N+1}^r c_{2i/m} \right) ,} \nonumber \\ h_{6r}^{(0)} (3)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N {\Big (b_{2i/m}^2 +c_{2i/m}^2 \Big )} , \nonumber \\ h_{6r}^{(0)} \left( i,j,l,1\right)= & {} \delta _{l+2N+1}^r b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,l\right) , \nonumber \\ h_{6r}^{(0)} \left( i,j,l,2\right)= & {} 2\delta _{l+3N+1}^r b_{2i/m} c_{2j/m} \Delta _2 \left( l,j,i\right) , \nonumber \\ h_{6r}^{(0)} \left( i,j,l,3\right)= & {} \delta _{l+2N+1}^r c_{2i/m} c_{2j/m} \Delta _2 \left( i,j,l\right) . \end{aligned}$$

(173)

The seventh term of the constant is

$$\begin{aligned} h_{7r}^{(0)} =\sum _{p=1}^3 {h_{7r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{7r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(174)

where

$$\begin{aligned} h_{7r}^{(0)} (1)= & {} 3\delta _0^r \big ({\dot{a}}_{10}^{(m)} \big )^{2}, \nonumber \\ h_{7r}^{(0)} (2)= & {} \frac{3}{2}\delta _0^r \sum _{i=1}^N {(P_{i/m}^2 +Q_{i/m}^2 )} , \nonumber \\ h_{7r}^{(0)} (3)= & {} 3{\dot{a}}_{10}^{(m)} \sum _{i=1}^N {(\delta _i^r P_{i/m} +\delta _{i+N}^r Q_{i/m} )} , \nonumber \\ h_{7r}^{(0)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+N}^r Q_{j/m} P_{l/m} +\delta _{j+N}^r Q_{i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _l^r Q_{i/m} Q_{j/m} \right] \Delta _2 \left( i,j,l\right) , \nonumber \\ h_{7r}^{(0)} \left( i,j,l,2\right)= & {} \left[ \delta _i^r P_{j/m} P_{l/m} +\delta _j^r P_{i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _l^r P_{i/m} P_{j/m} \right] \Delta _1 \left( i,j,l\right) . \end{aligned}$$

(175)

The eighth term of the constant is

$$\begin{aligned} h_{8r}^{(0)} =\sum _{p=1}^4 {h_{8r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{8r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(176)

where

$$\begin{aligned} h_{8r}^{(0)} (1)= & {} 2\delta _0^r {\dot{a}}_{10}^{(m)} a_{20}^{(m)} , \nonumber \\ h_{8r}^{(0)} (2)= & {} \sum _{i=1}^N {a_{20}^{(m)} (\delta _i^r P_{i/m} +\delta _{i+N}^r Q_{i/m} )} , \nonumber \\ h_{8r}^{(0)} (3)= & {} \sum _{i=1}^N {\delta _0^r \left( b_{2i/m} P_{i/m} +c_{2i/m} Q_{i/m} \right) } , \nonumber \\ h_{8r}^{(0)} (4)= & {} \sum _{i=1}^N {{\dot{a}}_{10}^{(m)} (\delta _i^r b_{(2)i/m} +\delta _{i+N}^r c_{(2)i/m} )} , \nonumber \\ h_{8r}^{(0)} \left( i,j,l,1\right)= & {} b_{2i/m} \left[ \delta _j^r P_{l/m} +\delta _l^r P_{j/m} \right] \Delta _1 \left( i,j,l\right) , \nonumber \\ h_{8r}^{(0)} \left( i,j,l,2\right)= & {} b_{2i/m} \left[ \delta _{j+N}^r Q_{l/m} +\delta _{l+N}^r Q_{j/m} \right] \Delta _2 \left( l,j,i\right) , \nonumber \\ h_{8r}^{(0)} \left( i,j,l,3\right)= & {} 2c_{2i/m} \left[ \delta _j^r Q_{l/m} +\delta _{j+N}^r P_{j/m} \right] \Delta _2 \left( i,l,j\right) ]. \nonumber \\ \end{aligned}$$

(177)

The ninth term of the constant is

$$\begin{aligned} h_{9r}^{(0)} =\sum _{p=1}^2 {h_{9r}^{(0)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{9r}^{(0)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(178)

where

$$\begin{aligned} h_{9r}^{(0)} (1)= & {} \delta _0^r \big (a_{20}^{(m)}\big )^{2}, \nonumber \\ h_{9r}^{(0)} (2)= & {} \frac{1}{2}\sum _{i=1}^N \big [2a_{20}^{(m)} \left( \delta _i^r b_{2i/m} +\delta _{i+N}^r c_{2i/m} \right) \nonumber \\&+\,\delta _0^r \big (b_{2i/m}^2 +c_{2i/m}^2 \big )\big ], \nonumber \\ h_{9r}^{(0)} \left( i,j,l,1\right)= & {} 2\delta _{l+3N+1}^r b_{2i/m} c_{2j/m} \Delta _2 \left( l,j,i\right) , \nonumber \\ h_{9r}^{(0)} \left( i,j,l,2\right)= & {} \delta _{l+2N+1}^r b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,l\right) , \nonumber \\ h_{9r}^{(0)} \left( i,j,l,3\right)= & {} c_{2i/m} c_{2j/m} \delta _{l+2N+1}^r \Delta _2 \left( i,j,l\right) . \end{aligned}$$

(179)

The tenth and eleventh terms of the constant are

$$\begin{aligned} h_{10r}^{(0)} =h_{11r}^{(0)} =0. \end{aligned}$$

(180)

For the derivatives of $f_\lambda ^{(c)}$ ($\lambda =1,2,\ldots ,11)$ with ${\dot{z}}_r$, the first term of cosine coefficients is

$$\begin{aligned} h_{1kr}^{(c)} =\sum _{p=1}^3 {h_{1kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{1kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(181)

where

$$\begin{aligned} h_{1kr}^{(c)} (1)= & {} 6\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} B_{k/m} +3\delta _{k+2N+1}^r ({\dot{a}}_{20}^{(m)} )^{2}, \nonumber \\ h_{1kr}^{(c)} (2)= & {} \frac{3}{2}\delta _0^r \sum _{i=1}^N\sum _{j=1}^N \left[ B_{i/m} B_{j/m} \Delta _1\left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} \Delta _2\left( i,j,k\right) \right] , \nonumber \\ h_{1kr}^{(c)} (3)= & {} \frac{3}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \Big [\delta _{i+2N+1}^r B_{j/m} \right. \nonumber \\&+\, \delta _{j+2N+1}^r B_{i/m} \Big ]\Delta _1 \left( i,j,k\right) \nonumber \\&\left. +\,\left[ \delta _{i+3N+1}^r C_{j/m} +\delta _{j+3N+1}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \right\} , \nonumber \end{aligned}$$

$$\begin{aligned} h_{1kr}^{(c)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+3N+1}^r C_{j/m} B_{l/m} +\delta _{j+3N+1}^r C_{i/m} B_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r C_{i/m} C_{j/m} \right] \Delta _4 \left( i,j,k,l\right) , \nonumber \\ h_{1kr}^{(c)} \left( i,j,l,2\right)= & {} \Big [\delta _{i+2N+1}^r B_{j/m} B_{l/m} +\,\delta _{j+2N+1}^r B_{i/m} B_{l/m} \nonumber \\&+\,\delta _{l+2N+1}^r B_{i/m} B_{j/m} \Big ]\Delta _3 \left( i,j,k,l\right) . \end{aligned}$$

(182)

The second term of cosine coefficients is

$$\begin{aligned} h_{2kr}^{(c)} =\sum _{p=1}^6 {h_{2kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{2kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(183)

where

$$\begin{aligned} h_{2kr}^{(c)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} P_{k/m} +\delta _k^r ({\dot{a}}_{20}^{(m)} )^{2}P_{k/m} , \nonumber \\ h_{2kr}^{(c)} (2)= & {} 2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} B_{k/m} +2\delta _0^r {\dot{a}}_{20}^{(m)} B_{k/m} \nonumber \\&+\,2\delta _{k+2N+1}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ h_{2kr}^{(c)} (3)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{2kr}^{(c)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r P_{j/m}\right. \nonumber \\&\left. +\,\delta _j^r B_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r Q_{j/m} +\delta _{j+N}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ h_{2kr}^{(c)} (5)= & {} \frac{1}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{2kr}^{(c)} (6)= & {} \;\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r B_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+2N+1}^r B_{i/m} \right] \Delta _1 \left( i,j,k\right) +\left[ \delta _{i+3N+1}^r C_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ h_{2kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+N}^r C_{j/m} P_{l/m} +\delta _{j+N}^r C_{i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r C_{i/m} C_{j/m} \right] \Delta _4 \left( i,j,k,l\right) , \nonumber \\ h_{2kr}^{(c)} \left( i,j,l,2\right)= & {} 2\left[ \delta _{i+2N+1}^r C_{j/m} Q_{l/m} +\delta _{j+3N+1}^r B_{i/m} Q_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r B_{i/m} C_{j/m} \right] \Delta _5 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(184)

The third term of cosine coefficients is

$$\begin{aligned} h_{3kr}^{(c)} =\sum _{p=1}^4 {h_{3kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{3kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(185)

where

$$\begin{aligned} h_{3kr}^{(c)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} B_{k/m} +2\delta _{k+2N+1}^r {\dot{a}}_{20}^{(m)} a_{20}^{(m)} +\nonumber \\&2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} b_{2k/m} , \nonumber \\ h_{3kr}^{(c)} (2)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \Big [\delta _{i+2N+1}^r B_{j/m} \right. \nonumber \\&+\,\delta _{j+2N+1}^r B_{i/m} \Big ]\Delta _1 \left( i,j,k\right) \nonumber \\&\left. +\,\left[ \delta _{i+3N+1}^r C_{j/m} +\delta _{j+3N+1}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \right\} , \nonumber \\ h_{3kr}^{(c)} (3)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{3kr}^{(c)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+2N+1}^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \end{aligned}$$

$$\begin{aligned} h_{3kr}^{(c)} \left( i,j,l,1\right)= & {} b_{2i/m} \left[ \delta _{j+2N+1}^r B_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r B_{j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{3kr}^{(c)} \left( i,j,l,2\right)= & {} \;b_{2i/m} \left[ \delta _{j+3N+1}^r C_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+3N+1}^r C_{j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{3kr}^{(c)} \left( i,j,l,3\right)= & {} 2c_{2i/m} \left[ \delta _{j+2N+1}^r C_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+3N+1}^r B_{j/m} \right] \Delta _6 \left( i,j,k,l\right) . \end{aligned}$$

(186)

The fourth term of cosine coefficients is

$$\begin{aligned} h_{4kr}^{(c)} =\sum _{p=1}^6 {h_{4kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{4kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(187)

where

$$\begin{aligned} h_{4kr}^{(c)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} P_{k/m}\nonumber \\&+\,2\delta _0^r {\dot{a}}_{20}^{(m)} P_{k/m} +2\delta _k^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} P_{k/m} , \nonumber \\ h_{4kr}^{(c)} (2)= & {} 2\delta _0^r {\dot{a}}_{10}^{(m)} B_{k/m} +\delta _{k+2N+1}^r \big ({\dot{a}}_{10}^{(m)} \big )^{2}B_{k/m} , \nonumber \\ h_{4kr}^{(c)} (3)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ P_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,Q_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{4kr}^{(c)} (4)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _i^r P_{j/m} +\,\delta _j^r P_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\,\left[ \delta _{i+N}^r Q_{j/m} +\delta _{j+N}^r Q_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \\ h_{4kr}^{(c)} (5)= & {} \delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,C_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{4kr}^{(c)} (6)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r P_{j/m} \right. \nonumber \\&\left. +\,\delta _j^r B_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&+\left[ \delta _{i+3N+1}^r Q_{j/m} +\delta _{j+N}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \}, \nonumber \end{aligned}$$

$$\begin{aligned} h_{4kr}^{(c)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r P_{j/m} P_{l/m} +\delta _j^r B_{i/m} P_{l/m} \right. \nonumber \\&\left. +\,\delta _l^r B_{i/m} P_{j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{4kr}^{(c)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+2N+1}^r Q_{j/m} Q_{l/m} +\delta _{j+N}^r B_{i/m} Q_{l/m}\right. \nonumber \\&\left. +\,\delta _{l+N}^r B_{i/m} Q_{j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{4kr}^{(c)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r P_{j/m} Q_{l/m} +\delta _j^r C_{i/m} Q_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r C_{i/m} P_{j/m} \right] \Delta _6 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(188)

The fifth term of cosine coefficients is

$$\begin{aligned} h_{5kr}^{(c)} =\sum _{p=1}^8 {h_{5kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^4 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{5kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(189)

where

$$\begin{aligned} h_{5kr}^{(c)} (1)= & {} \delta _{2N+1}^r a_{20}^{(m)} P_{k/m} +\delta _k^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ h_{5kr}^{(c)} (2)= & {} \delta _0^r a_{20}^{(m)} B_{k/m} +\delta _{k+2N+1}^r a_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ h_{5kr}^{(c)} (3)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \left[ \delta _{i+2N+1}^r P_{j/m}\right. \right. \nonumber \\&\left. +\,\delta _j^r B_{i/m} \right] \Delta _1 \left( i,j,k\right) \nonumber \\&\left. +\,\left[ \delta _{i+3N+1}^r Q_{j/m} +\delta _{j+N}^r C_{i/m} \right] \Delta _2 \left( i,j,k\right) \right\} , \nonumber \\ h_{5kr}^{(c)} (4)= & {} \delta _{2N+1}^r {\dot{a}}_{10}^{(m)} b_{2k/m} +\delta _0^r {\dot{a}}_{20}^{(m)} b_{2k/m} , \nonumber \\ h_{5kr}^{(c)} (5)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{5kr}^{(c)} (6)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _j^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+N}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{5kr}^{(c)} (7)= & {} \;\frac{1}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} B_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} C_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{5kr}^{(c)} (8)= & {} \;\frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+2N+1}^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ \end{aligned}$$

$$\begin{aligned} h_{5kr}^{(c)} \left( i,j,l,1\right)= & {} b_{2i/m} \left[ \delta _{j+2N+1}^r P_{l/m} +\delta _l^r B_{j/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(c)} \left( i,j,l,2\right)= & {} b_{2i/m} \left[ \delta _{j+3N+1}^r Q_{l/m} +\delta _{l+N}^r C_{j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(c)} \left( i,j,l,3\right)= & {} c_{2i/m} \left[ \delta _{j+2N+1}^r Q_{l/m} +\delta _{l+N}^r B_{j/m} \right] \Delta _6 \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(c)} \left( i,j,l,4\right)= & {} c_{2i/m} \left[ \delta _{j+3N+1}^r P_{l/m} +\delta _l^r C_{j/m} \right] \Delta _4 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(190)

The sixth term of cosine coefficients is

$$\begin{aligned} h_{6kr}^{(c)} =\sum _{p=1}^3 {h_{6kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{6kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(191)

where

$$\begin{aligned} h_{6kr}^{(c)} (1)= & {} \delta _{k+2N+1}^r \big (a_{20}^{(m)}\big )^{2}+2\delta _{2N+1}^r a_{20}^{(m)} b_{2k/m} , \nonumber \\ h_{6kr}^{(c)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+2N+1}^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+N}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{6kr}^{(c)} (3)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} c_{2j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{6kr}^{(c)} \left( i,j,l,1\right)= & {} \delta _{l+2N+1}^r b_{2i/m} b_{2j/m} \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{6kr}^{(c)} \left( i,j,l,2\right)= & {} \delta _{l+3N+1}^r b_{2i/m} c_{2j/m} \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{6kr}^{(c)} \left( i,j,l,3\right)= & {} \delta _{l+2N+1}^r c_{2i/m} c_{2j/m} \Delta _4 \left( i,j,k,l\right) . \end{aligned}$$

(192)

The seventh term of cosine coefficients is

$$\begin{aligned} h_{7kr}^{(c)} =\sum _{p=1}^3 {h_{7kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{7kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(193)

where

$$\begin{aligned} h_{7kr}^{(c)} (1)= & {} 6\delta _0^r {\dot{a}}_{10}^{(m)} P_{k/m} +3\delta _k^r \big ({\dot{a}}_{10}^{(m)} \big ){ }^2, \nonumber \\ h_{7kr}^{(c)} (2)= & {} \frac{3}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ P_{j/m} P_{k/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,Q_{i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{7kr}^{(c)} (3)= & {} \frac{3}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\Big [\delta _i^r P_{j/m}+\,\delta _j^r P_{i/m} \Big ]\Delta _1 \left( i,j,k\right) \nonumber \\&+\,\Big [\delta _{i+N}^r Q_{j/m} +\delta _{j+N}^r Q_{i/m} \Big ]\Delta _2 \left( i,j,k\right) \}, \nonumber \\ h_{7kr}^{(c)} \left( i,j,l,1\right)= & {} 3\Big [\delta _{i+N}^r Q_{j/m} P_{l/m} +\delta _{j+N}^r Q_{i/m} P_{l/m} \nonumber \\&+\,\delta _l^r Q_{i/m} Q_{j/m} \Big ]\Delta _4 \left( i,j,k,l\right) , \nonumber \\ h_{7kr}^{(c)} \left( i,j,l,2\right)= & {} \Big [\delta _i^r P_{j/m} P_{l/m} +\delta _j^r P_{i/m} P_{l/m} \nonumber \\&+\,\delta _l^r P_{i/m} P_{j/m} \Big ]\Delta _3 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(194)

The eighth term of cosine coefficients is

$$\begin{aligned} h_{8kr}^{(c)} =\sum _{p=1}^4 {h_{8kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{8kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(195)

where

$$\begin{aligned} h_{8kr}^{(c)} (1)= & {} 2\delta _0^r {\dot{a}}_{20}^{(m)} P_{k/m} +2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} P_{k/m} \nonumber \\&+\,2\delta _{k+2N+1}^r {\dot{a}}_{10}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ h_{8kr}^{(c)} (2)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \left[ \delta _i^r P_{j/m} +\delta _j^r P_{i/m} \right] \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\left[ \delta _{i+N}^r Q_{j/m} +\delta _{j+N}^r Q_{i/m} \right] \Delta _2 \left( i,j,k\right) \right\} , \nonumber \\ h_{8kr}^{(c)} (3)= & {} \delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} P_{j/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,c_{2i/m} Q_{j/m} \Delta _2 \left( i,j,k\right) \right] ,\;\; \nonumber \\ h_{8kr}^{(c)} (4)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _j^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+N}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \end{aligned}$$

$$\begin{aligned} h_{8kr}^{(c)} \left( i,j,l,1\right)= & {} b_{2i/m} \left[ \delta _j^r P_{j/m} +\delta _l^r P_{l/m} \right] \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{8kr}^{(c)} \left( i,j,l,2\right)= & {} b_{2i/m} \left[ \delta _{j+N}^r Q_{l/m} +\delta _{l+N}^r Q_{j/m} \right] \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{8kr}^{(c)} \left( i,j,l,3\right)= & {} 2c_{2i/m} \left[ \delta _j^r Q_{l/m} +\delta _{l+N}^r P_{j/m} \right] \Delta _6 \left( i,j,k,l\right) .\nonumber \\ \end{aligned}$$

(196)

The ninth term of cosine coefficients is

$$\begin{aligned} h_{9kr}^{(c)} =\sum _{p=1}^3 {h_{9kr}^{(c)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{9kr}^{(c)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(197)

where

$$\begin{aligned} h_{9kr}^{(c)} (1)= & {} \delta _k^r \big (a_{20}^{(m)}\big )^{2}+2\delta _0^r a_{20}^{(m)} b_{2k/m} , \nonumber \\ h_{9kr}^{(c)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _j^r b_{2i/m} \Delta _1 \left( i,j,k\right) \right. \nonumber \\&\left. +\,\delta _{j+N}^r c_{2i/m} \Delta _2 \left( i,j,k\right) \right] , \nonumber \\ h_{9kr}^{(c)} (3)= & {} \frac{1}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \Big [b_{2i/m} b_{2j/m} \Delta _1 \left( i,j,k\right) \nonumber \\&+\,c_{2i/m} c_{2j/m} \Delta _2 \left( i,j,k\right) \Big ] , \nonumber \end{aligned}$$

$$\begin{aligned} h_{9kr}^{(c)} \left( i,j,l,1\right)= & {} \delta _l^r b_{2i/m} b_{2j/m} \Delta _3 \left( i,j,k,l\right) , \nonumber \\ h_{9kr}^{(c)} \left( i,j,l,2\right)= & {} 2\delta _{l+N}^r b_{2i/m} c_{2j/m} \Delta _5 \left( i,j,k,l\right) , \nonumber \\ h_{9kr}^{(c)} \left( i,j,l,3\right)= & {} \delta _l^r c_{2i/m} c_{2j/m} \Delta _4 \left( i,j,k,l\right) . \end{aligned}$$

(198)

The tenth and eleventh terms of cosine coefficients are

$$\begin{aligned} \frac{\partial f_{10k}^{(c)} }{\partial {\dot{z}}_r }=\frac{\partial f_{11k}^{(c)} }{\partial {\dot{z}}_r }=0 \end{aligned}$$

(199)

For the derivatives of $f_\lambda ^{(c)}$ ($\lambda =1,2,\ldots ,11)$ with ${\dot{z}}_r$, the first term of sine coefficients is

$$\begin{aligned} h_{1kr}^{(s)} =\sum _{p=1}^3 {h_{1kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{1kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(200)

where

$$\begin{aligned} h_{1kr}^{(s)} (1)= & {} 6\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} C_{k/m} +3\delta _{k+3N+1}^r ({\dot{a}}_{20}^{(m)} )^{2}C_{k/m} , \nonumber \\ h_{1kr}^{(s)} (2)= & {} 3\delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} \Delta _2 \left( k,j,i\right) } } , \nonumber \\ h_{1kr}^{(s)} (3)= & {} 3{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{i+2N+1}^r B_{i/m} C_{j/m}\right. \nonumber \\&\left. +\,\delta _{j+3N+1}^r B_{i/m} C_{j/m} \right] \Delta _2 \left( k,j,i\right) ,\nonumber \\ h_{1kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _{i+2N+1}^r C_{j/m} B_{l/m} +\delta _{j+3N+1}^r B_{i/m} B_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+2N+1}^r B_{i/m} C_{j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ h_{1kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+3N+1}^r C_{j/m} C_{l/m} +\delta _{j+3N+1}^r C_{i/m} C_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+3N+1}^r C_{i/m} C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(201)

The second term of sine coefficients is

$$\begin{aligned} h_{2kr}^{(s)} =\sum _{p=1}^6 {h_{2kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{2kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(202)

where

$$\begin{aligned} h_{2kr}^{(s)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} Q_{k/m} +2\delta _{k+N}^r ({\dot{a}}_{20}^{(m)} )^{2}, \nonumber \\ h_{2kr}^{(s)} (2)= & {} 2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} C_{k/m} +2\delta _0^r {\dot{a}}_{20}^{(m)} C_{k/m} \nonumber \\&+2\delta _{k+3N+1}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ h_{2kr}^{(s)} (3)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\,C_{i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{2kr}^{(s)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \{\left[ \delta _{i+2N+1}^r Q_{j/m} \right. \nonumber \\&\left. +\,\delta _{j+N}^r B_{i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&+\,\left[ \delta _{i+3N+1}^r P_{j/m} +\delta _j^r C_{i/m} \right] \Delta _2 \left( i,k,j\right) \}, \nonumber \\ h_{2kr}^{(s)} (5)= & {} \delta _0^r \sum _{i=1}^N {\sum _{j=1}^N {B_{i/m} C_{j/m} \Delta _2 \left( k,j,i\right) ,} } \nonumber \\ h_{2kr}^{(s)} (6)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big [\delta _{i+2N+1}^r C_{j/m} \nonumber \\&+\, \delta _{j+3N+1}^r B_{i/m} \Big ]\Delta _2 \left( k,j,i\right) ,\; \nonumber \end{aligned}$$

$$\begin{aligned} h_{2kr}^{(s)} \left( i,j,l,1\right)= & {} \left[ \delta _{i+2N+1}^r B_{j/m} Q_{l/m} +\delta _{j+2N+1}^r B_{i/m} Q_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r B_{i/m} B_{j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{2kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+3N+1}^r C_{j/m} Q_{l/m} +\delta _{j+3N+1}^r C_{i/m} C_{j/m} Q_{l/m} \right. \nonumber \\&\left. +\,\delta _{l+N}^r C_{i/m} C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) , \nonumber \\ h_{2kr}^{(s)} \left( i,j,l,3\right)= & {} 2[\delta _{i+2N+1}^r C_{j/m} P_{l/m} +\delta _{j+3N+1}^r B_{i/m} P_{l/m} \nonumber \\&+\,\delta _l^r B_{i/m} C_{j/m} ]\Delta _7 \left( i,j,k,l\right) . \end{aligned}$$

(203)

The third term of sine coefficients is

$$\begin{aligned} h_{3kr}^{(s)} =\sum _{p=1}^5 {h_{3kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{3kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(204)

where

$$\begin{aligned} h_{3kr}^{(s)} (1)= & {} 2\delta _{2N+1}^r a_{20}^{(m)} C_{k/m} +2\delta _{k+3N+1}^r {\dot{a}}_{20}^{(m)} a_{20}^{(m)} , \nonumber \\ h_{3kr}^{(s)} (2)= & {} 2\delta _{2N+1}^r {\dot{a}}_{20}^{(m)} c_{2k/m} , \nonumber \\ h_{3kr}^{(s)} (3)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big [\delta _{i+2N+1}^r C_{j/m} \nonumber \\&+\, \delta _{j+3N+1}^r B_{i/m} \Big ]\Delta _2 \left( k,j,i\right) ,\nonumber \\ h_{3kr}^{(s)} (4)= & {} \delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\, c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{3kr}^{(s)} (5)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} \delta _{j+3N+1}^r \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\delta _{j+2N+1}^r c_{2i/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{3kr}^{(s)} \left( i,j,l,1\right)= & {} 2b_{2i/m} \Big [\delta _{j+2N+1}^r C_{l/m} \nonumber \\&+\, \delta _{l+3N+1}^r B_{j/m} \Big ]\Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{3kr}^{(s)} \left( i,j,l,2\right)= & {} c_{2i/m} \Big [\delta _{j+2N+1}^r B_{l/m} \nonumber \\&+\, \delta _{l+2N+1}^r B_{j/m} \Big ]\Delta _{10} \left( i,j,k,l\right) , \nonumber \\ h_{3kr}^{(s)} \left( i,j,l,3\right)= & {} c_{2i/m} \left[ \delta _{j+3N+1}^r C_{l/m}\right. \nonumber \\&\left. +\, \delta _{l+3N+1}^r C_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(205)

The fourth term of sine coefficients is

$$\begin{aligned} h_{4kr}^{(s)} =\sum _{p=1}^6 {h_{4kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{4kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(206)

where

$$\begin{aligned} h_{4kr}^{(s)} (1)= & {} 2\delta _{2N+1}^r {\dot{a}}_{10}^{(m)} Q_{k/m} +2\delta _0^r {\dot{a}}_{20}^{(m)} Q_{k/m} , \nonumber \\ h_{4kr}^{(s)} (2)= & {} 2\delta _{k+N}^r {\dot{a}}_{20}^{(m)} {\dot{a}}_{10}^{(m)} +2\delta _0^r {\dot{a}}_{10}^{(m)} C_{k/m}\nonumber \\&+\delta _{k+3N+1}^r \big ({\dot{a}}_{10}^{(m)} \big )^{2}, \nonumber \\ h_{4kr}^{(s)} (3)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {P_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) } } , \nonumber \\ h_{4kr}^{(s)} (4)= & {} {\dot{a}}_{20}^{(m)} \sum _{i=1}^N {\sum _{j=1}^N {[\delta _i^r Q_{j/m} +\delta _{j+N}^r P_{i/m} ]\Delta _2 \left( k,j,i\right) } } , \nonumber \\ h_{4kr}^{(s)} (5)= & {} \delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ B_{i/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +C_{i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{4kr}^{(s)} (6)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big \{\left[ \delta _{i+2N+1}^r Q_{j/m} \right. \nonumber \\&\left. +\delta _{j+N}^r B_{i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&\left. +\left[ \delta _{i+3N+1}^r P_{j/m} +\delta _j^r C_{i/m} \right] \Delta _2 \left( i,k,j\right) \right\} , \nonumber \end{aligned}$$

$$\begin{aligned} h_{4kr}^{(s)} \left( i,j,l,1\right)= & {} 2\left[ \delta _{i+2N+1}^r P_{j/m} Q_{l/m} +\delta _j^r B_{i/m} Q_{l/m}\right. \nonumber \\&\left. +\delta _{l+N}^r B_{i/m} P_{j/m} \right] \Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{4kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+3N+1}^r P_{j/m} P_{l/m} +\delta _j^r C_{i/m} P_{l/m} \right. \nonumber \\&\left. +\delta _l^r C_{i/m} P_{j/m} \right] \Delta _{10} \left( i,j,k,l\right) , \nonumber \\ h_{4kr}^{(s)} \left( i,j,l,3\right)= & {} \left[ \delta _{i+3N+1}^r Q_{j/m} Q_{l/m} +\delta _{j+N}^r C_{i/m} Q_{l/m}\right. \nonumber \\&\left. +\delta _{l+N}^r C_{i/m} Q_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(207)

The fifth term of sine coefficients is

$$\begin{aligned} h_{5kr}^{(s)} =\sum _{p=1}^8 {h_{5kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^4 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{5kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(208)

where

$$\begin{aligned} h_{5kr}^{(s)} (1)= & {} \delta _{2N+1}^r a_{20}^{(m)} Q_{k/m} +\delta _{k+N}^r a_{20}^{(m)} {\dot{a}}_{20}^{(m)} , \nonumber \\ h_{5kr}^{(s)} (2)= & {} \delta _0^r a_{20}^{(m)} C_{k/m} +\delta _{k+3N+1}^r a_{20}^{(m)} {\dot{a}}_{10}^{(m)} , \nonumber \\ h_{5kr}^{(s)} (3)= & {} \frac{1}{2}a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left\{ \left[ \delta _{i+2N+1}^r Q_{j/m} \right. \right. \nonumber \\&\left. +\delta _{j+N}^r B_{i/m} \right] \Delta _2 \left( k,j,i\right) \nonumber \\&\left. +\left[ \delta _{i+3N+1}^r P_{j/m} +\delta _j^r C_{i/m} \right] \Delta _2 \left( i,k,j\right) \right\} , \nonumber \\ h_{5kr}^{(s)} (4)= & {} \delta _{2N+1}^r {\dot{a}}_{10}^{(m)} c_{2k/m} +\delta _0^r {\dot{a}}_{20}^{(m)} c_{2k/m} , \nonumber \\ h_{5kr}^{(s)} (5)= & {} \frac{1}{2}\delta _{2N+1}^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2j/m} Q_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +c_{2i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{5kr}^{(s)} (6)= & {} \frac{1}{2}{\dot{a}}_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+N}^r b_{2j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\delta _j^r c_{2i/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{5kr}^{(s)} (7)= & {} \frac{1}{2}\delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2i/m} C_{j/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +c_{2i/m} B_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{5kr}^{(s)} (8)= & {} \frac{1}{2}{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+3N+1}^r b_{2i/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\delta _{j+2N+1}^r c_{2i/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{5kr}^{(s)} \left( i,j,l,1\right)= & {} b_{2i/m} \Big [\delta _{j+2N+1}^r Q_{l/m} \nonumber \\&+\, \delta _{l+N}^r B_{j/m} \Big ]\Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(s)} \left( i,j,l,2\right)= & {} b_{2i/m} \Big [\delta _{j+3N+1}^r P_{l/m} \nonumber \\&+\, \delta _l^r C_{j/m} \Big ]\Delta _7 \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(s)} \left( i,j,l,3\right)= & {} c_{2i/m} \Big [\delta _{j+2N+1}^r P_{l/m} \nonumber \\&+\, \delta _l^r B_{j/m} \Big ]\Delta _{10} \left( i,j,k,l\right) , \nonumber \\ h_{5kr}^{(s)} \left( i,j,l,4\right)= & {} c_{2i/m} \Big [\delta _{j+3N+1}^r Q_{l/m} \nonumber \\&+\, \delta _{l+N}^r C_{j/m}\Big ]\Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(209)

The sixth term of sine coefficients is

$$\begin{aligned} h_{6kr}^{(s)} =\sum _{p=1}^3 {h_{6kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{6kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(210)

where

$$\begin{aligned} h_{6kr}^{(s)} (1)= & {} \delta _{k+3N+1}^r \left( a_{20}^{(m)}\right) ^{2}+2\delta _{k+2N+1}^r a_{20}^{(m)} c_{2k/m} , \nonumber \\ h_{6kr}^{(s)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+2N+1}^r c_{2i/m} \Delta _2 \left( i,k,j\right) \right. \nonumber \\&\left. +\delta _{j+3N+1}^r b_{2i/m} \Delta _2 \left( k,j,i\right) \right] , \nonumber \\ h_{6kr}^{(s)} (3)= & {} \delta _{2N+1}^r \sum _{i=1}^N {\sum _{j=1}^N {c_{2i/m} b_{2j/m} \Delta _2 \left( i,k,j\right) ,} } \; \nonumber \\ h_{6kr}^{(s)} \left( i,j,l,1\right)= & {} \delta _{l+3N+1}^r b_{2i/m} b_{2j/m} \Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{6kr}^{(s)} \left( i,j,l,2\right)= & {} 2\delta _{l+2N+1}^r b_{2i/m} c_{2j/m} \Delta _7 \left( i,j,k,l\right) , \nonumber \\ h_{6kr}^{(s)} \left( i,j,l,3\right)= & {} \delta _{l+3N+1}^r c_{2i/m} c_{2j/m} \Delta _9 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(211)

The seventh term of sine coefficients is

$$\begin{aligned} h_{7kr}^{(s)} =\sum _{p=1}^3 {h_{7kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^2 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{7kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(212)

where

$$\begin{aligned} h_{7kr}^{(s)} (1)= & {} 6\delta _0^r {\dot{a}}_{10}^{(m)} Q_{k/m} +3\delta _{k+N}^r \big ({\dot{a}}_{10}^{(m)} \big )^{2}, \nonumber \\ h_{7kr}^{(s)} (2)= & {} 3\delta _0^r \sum _{i=1}^N {\sum _{j=1}^N {P_{i/m} Q_{j/m} } \Delta _2 \left( k,j,i\right) } , \nonumber \\ h_{7kr}^{(s)} (3)= & {} 3{\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N {[\delta _i^r Q_{j/m} +\delta _{j+N}^r P_{i/m} ]}\nonumber \\&\Delta _2 \left( k,j,i\right) ,\nonumber \\ h_{7kr}^{(s)} \left( i,j,l,1\right)= & {} 3\left[ \delta _i^r Q_{j/m} P_{l/m} +\delta _{j+N}^r P_{i/m} P_{l/m} \right. \nonumber \\&\left. +\delta _l^r P_{i/m} Q_{j/m} \right] \Delta _7 \left( i,j,k,l\right) , \nonumber \\ h_{7kr}^{(s)} \left( i,j,l,2\right)= & {} \left[ \delta _{i+N}^r Q_{j/m} Q_{l/m} +\delta _{j+N}^r Q_{i/m} Q_{l/m} \right. \nonumber \\&\left. +\delta _{l+N}^r Q_{i/m} Q_{j/m} \right] \Delta _8 \left( i,j,k,l\right) . \nonumber \\ \end{aligned}$$

(213)

The eighth term of sine coefficients is

$$\begin{aligned} h_{8kr}^{(s)} =\sum _{p=1}^4 {h_{8kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{8kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(214)

where

$$\begin{aligned} h_{8kr}^{(s)} (1)= & {} 2\delta _0^r a_{20}^{(m)} Q_{k/m} +2\delta _{k+N}^r {\dot{a}}_{10}^{(m)} a_{20}^{(m)} , \nonumber \\ h_{8kr}^{(s)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big [\delta _i^r Q_{j/m} +\,\delta _{j+N}^r P_{i/m} \Big ]\Delta _2 \left( k,j,i\right) , \nonumber \\ h_{8kr}^{(s)} (3)= & {} \delta _0^r \sum _{i=1}^N \sum _{j=1}^N \left[ b_{2j/m} Q_{k/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +c_{2i/m} P_{j/m} \Delta _2 \left( i,k,j\right) \right] , \nonumber \\ h_{8kr}^{(s)} (4)= & {} {\dot{a}}_{10}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \left[ \delta _{j+N}^r b_{2i/m} \Delta _2 \left( k,j,i\right) \right. \nonumber \\&\left. +\delta _j^r c_{2i/m} \Delta _2 \left( i,k,j\right) \right] +2\delta _0^r {\dot{a}}_{10}^{(m)} c_{2k/m} , \nonumber \end{aligned}$$

$$\begin{aligned} h_{8kr}^{(s)} \left( i,j,l,1\right)= & {} 2b_{2i/m} \Big [\delta _j^r Q_{l/m} \nonumber \\&+\, \delta _{l+N}^r P_{j/m} \Big ]\Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{8kr}^{(s)} \left( i,j,l,2\right)= & {} c_{2i/m} \Big [\delta _j^r P_{l/m} \nonumber \\&+\, \delta _l^r P_{j/m} \Big ]\Delta _{10} \left( i,j,k,l\right) , \nonumber \\ h_{8kr}^{(s)} \left( i,j,l,3\right)= & {} c_{2i/m} \Big [\delta _{j+N}^r Q_{l/m} \nonumber \\&+\, \delta _{l+N}^r Q_{j/m} \Big ]\Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(215)

The ninth term of sine coefficients is

$$\begin{aligned} h_{9kr}^{(s)} =\sum _{p=1}^3 {h_{9kr}^{(s)} (p)} +\;\frac{1}{4}\sum _{q=1}^3 {\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {h_{9kr}^{(s)} \left( i,j,l,q\right) } } } }\nonumber \\ \end{aligned}$$

(216)

where

$$\begin{aligned} h_{9kr}^{(s)} (1)= & {} \delta _{k+N}^r \big (a_{20}^{(m)}\big )^{2}+2\delta _0^r a_{20}^{(m)} c_{2k/m} , \nonumber \\ h_{9kr}^{(s)} (2)= & {} a_{20}^{(m)} \sum _{i=1}^N \sum _{j=1}^N \Big [\delta _{j+N}^r b_{2i/m} \Delta _2 \left( k,j,i\right) \nonumber \\&+\delta _j^r c_{2i/m} \Delta _2 \left( i,k,j\right) \Big ] , \nonumber \\ h_{9kr}^{(s)} (3)= & {} \delta _0^r \sum _{i=1}^N {\sum _{j=1}^N {b_{2i/m} c_{2j/m} \Delta _2 \left( k,j,i\right) } } , \nonumber \\ h_{9kr}^{(s)} \left( i,j,l,1\right)= & {} 2b_{2i/m} c_{2j/m} \delta _l^r \Delta _7 \left( i,j,k,l\right) , \nonumber \\ h_{9kr}^{(s)} \left( i,j,l,2\right)= & {} \delta _{l+N}^r b_{2i/m} b_{2j/m} \Delta _9 \left( i,j,k,l\right) , \nonumber \\ h_{9kr}^{(s)} \left( i,j,l,3\right)= & {} \delta _{l+N}^r c_{2i/m} c_{2j/m} \Delta _8 \left( i,j,k,l\right) . \end{aligned}$$

(217)

The tenth and eleventh terms of sine coefficients are

$$\begin{aligned} \frac{\partial f_{10k}^{(s)} }{\partial {\dot{z}}_r }=\frac{\partial f_{11k}^{(s)} }{\partial {\dot{z}}_r }=0. \end{aligned}$$

(218)